WORKING 2003 /9/ PAPER SERIES - cnb.cz

WORKINGPAPER SERIES

/9/2003

WORKING PAPER SERIES

Credit Risk, Systemic Uncertainties and Economic CapitalRequirements for an Artificial Bank Loan Portfolio

Alexis DervizNarcisa Kadlčáková

Lucie Kobzová

9/2003

THE WORKING PAPER SERIES OF THE CZECH NATIONAL BANK

The Working Paper series of the Czech National Bank (CNB) is intended to disseminate theresults of the CNB-coordinated research projects, as well as other research activities of boththe staff of the CNB and colaborating outside contributors. This series supersedes previouslyissued research series (Institute of Economics of the CNB; Monetary Department of theCNB). All Working Papers are refereed internationally and the views expressed therein arethose of the authors and do not necessarily reflect the official views of the CNB.

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© Czech National Bank, December 2003Alexis Derviz, Narcisa Kadlčáková, Lucie Kobzová

Credit Risk, Systemic Uncertainties and Economic Capital Requirementsfor an Artificial Bank Loan Portfolio

Alexis Derviz, Narcisa Kadlčáková, Lucie Kobzová*

Abstract

This paper analyses the impact of different credit risk-based capital requirementimplementations on banks’ need for capital. The capital requirements for an artificiallyconstructed risky loan portfolio are calculated by applying the BIS approach, the twowidespread commercial risk-measurement models, CreditMetrics and CreditRisk+, and, finally,an original synthetic model similar to KMV. In the first three cases we closely follow themethodologies proposed by the regulatory or credit risk models. Economic capital requirementsfor the latter are obtained by means of Monte Carlo simulations. In the context of CreditMetrics,we additionally perform a Monte Carlo-based stress testing of the monetary policy changesreflected in the term structure of interest rates. Our model of KMV type combines the elementsof the structural and the reduced-form methods of risky debt pricing, and the possibilities of itsnumerical solution are outlined.

JEL Codes: G21, G28, G33.Keywords: credit risk, economic capital, market risk, New Basel Capital Accord, systemic

uncertainty.

* Alexis Derviz: The Czech National Bank, Monetary and Statistical Departmant, and Insitute for InformationTheory and Automation, Prague, E-mail:[email protected].; Narcisa Kadlčáková: The Czech NationalBank, Monetary and Statistical Departmant, E-mail:[email protected]; Lucie Kobzová: The CzechNational Bank, Banking Supervision Departmant, E-mail:[email protected] project supported by the Czech National Bank, Research Project No.17/2001.

2 Alexis Derviz, Narcisa Kadlčáková, Lucie Kobzová

Nontechnical Summary

Central banks need to understand the mechanisms of loan valuation and prudential capitaldetermination used by commercial banks for two main reasons. First, they are responsible fororderly operation of the banking sector. This aspect of financial stability is strongly influenced bythe credit risk management practices of the banks under their jurisdiction. Currently, thedevelopment of global standards in this area, summarized by the New Basel Capital Accord, itsextensions and amendments, is exerting systematic pressure on banks to modernize their creditrisk management procedures, and the central bank regulatory role is evolving accordingly.Second, domestic credit creation in the banking sector impacts decisively on the transmission ofmonetary policy, which is the core central bank responsibility. Lending to the real economy andthe balance sheet situation of banks are much dependent upon their attitude towards assessing, andprovision-taking for, risky loans.

The present paper investigates the impact of different credit risk-based capital requirementimplementations on banks’ need for capital. By constructing an artificial bank loan portfolio thatmimics key aggregate features of Czech corporate and household borrowing, we gauge thelending-bank economic capital requirements generated by the borrower side of the Czecheconomy. We start by calculating the regulatory requirement according to the BIS approach, andproceed by applying to the same portfolio the two widespread commercial credit-risk models:CreditMetrics and CreditRisk+. Economic capital for the former is obtained by means of MonteCarlo simulations. In conducting them, we took into account the monetary policy uncertaintiesreflected in random movements of the Czech term structure and investigated a number ofscenarios of key interest rate changes and their transmission along the yield curve. The well-known shortcomings of the conventional credit risk models have to do with non-tradability ofmajor parts of the loan portfolio, the counter-intuitive dependence of the loan value on marketinterest rates, as well as the pro-cyclical behavior of economic capital implied by the existingmodels. It was necessary to analyze the banks’ way of dealing with both problems realistically.Therefore, we calculated economic capital requirements for the same artificial portfolio in ourown credit risk model, which extends the well-known KMV framework and addresses the namedissues of market incompleteness and pro-cyclicity.

The main findings of the paper and its contribution to the literature are the following:1. Risky debt valuation by the traditional asset pricing methods currently in use by the banking

industry tends to generate higher loan values and reduce economic capital requirements,compared to other possible regulatory and model-based risk measurement methods. Therefore,the regulator may see an effort on the part of the banks to treat different parts of loans on theirbalances differently in terms of economic capital. The difference will go in the direction ofreducing capital allocation (and specializing collateral requirements) in those segments of theloan portfolio that exhibit strong correlation with traded risks.

2. Those methods of credit risk measurement which explicitly deal with market incompleteness(i.e. the lack of market valuation of both the loan itself and the assets of the obligor) lead to abetter recognition of the role of the business cycle and other systemic macroeconomic factorsin economic capital determination. Therefore, the regulator should encourage the use ofmethods that allow for counter-cyclical adjustments by banks of pro-cyclically biased risk-management procedures.

Credit Risk, Systemic Uncertainties and Economic Capital for an Artificial Bank Loan Portfolio 3

1. Introduction

Regular assessments of the default risk of bank clients and estimations of credit risk at theportfolio level are becoming a necessity for banks in their daily operations. The design of optimallending contracts and the need to conform to new regulatory trends constitute at least two reasonswhy banks have to pay closer attention to quantitative methods for assessing the credit risk oftheir clients. While primarily designed for use in commercial banks, credit risk models haverecently started to attract the attention of other groups of economic professionals. It is thesupervisory function of central banks that is mostly triggering the interest in examining credit riskmodels in this environment. Beside that, an overall assessment of the creditworthiness of domesticfirms has implications for the conduct of monetary policy. These and other reasons have promptedseveral central banks in Europe to develop and implement their own models for monitoring thefinancial situation of domestic firms and the lending performance of domestic banks1.

The objective of the present paper is to develop an assessment technique for analyzing the impactof different credit risk-based capital requirement determination methods on the potential needs forcapital in the Czech banking sector. For this purpose, we apply the discussed methods to capitalrequirement calculations for an artificially constructed risky loan portfolio. The latter reflects anumber of prominent features of Czech non-financial borrowers.

When defining the creditworthiness characteristics of the tested loan portfolio, we apply Moody’smethod for rating private firms. To determine capital requirements for this portfolio, we use theNew Basel Capital Accord (NBCA) and the CreditMetrics and CreditRisk+ models. In the contextof CreditMetrics, we are able to conduct stress testing to gauge the impact of interest rateuncertainty (e.g. caused by changes in monetary policy and different reactions of the yield curveto these changes) on economic capital calculations. In addition, we describe an independent debtvaluation model of KMV type and outline the techniques for its numerical implementation. Theproposed model has a substantial advantage over the previously mentioned ones in that itaddresses three key problems of credit risk modeling. Namely, this model, although remaining inthe KMV line of analysis,

• incorporates macroeconomic systemic factors, such as position in the business cycle, interestrate and exchange rate volatility and the monetary policy stance, when deriving a valuation ofbank lending risks,

• combines the features of structural and reduced-form models of debt valuation,• offers a framework for assessing the influence of market risk factors on credit risk in a bank

loan portfolio.

In the latter respect, our model advances towards an integrated financial risk assessmentmethodology, which has recently been called for in the risk analysis literature (see, for instance,Barnhill and Maxwell, 2002, or Hou, 2002).

The principal output of the paper is a comparative analysis of the predictions of these modelswhen applied to an artificially created loan portfolio constructed using Czech data. Another 1 Rating systems and creditworthiness-assessment models for firms have been developed, among others, by thecentral banks of France, Germany, Italy, Austria and the UK.


important contribution is a demonstration, even if in an incipient manner, of the way in whichscarce and usually unavailable variables can be estimated or proxied to obtain the inputs requiredby the credit risk models. One often-mentioned drawback of credit risk modeling is the difficultywith which the credit risk analyst can access the required input data. This problem was alsopresent in our case. Although overcome, the data problem has had negative implications for therobustness of our results. Thus, from this perspective we have to look at the paper’s findings withcaution. However, the insight into credit risk modeling that is offered here can be extended at alater stage when more data is available. We also hope that our findings may be of use to bankingsupervisors when these issues become a matter of regulatory practice.

Although credit risk models often prove useful for other purposes, their main merit rests inestimating the capital level that banks have to maintain over the given risk horizon. The outcomeis called regulatory capital in regulatory terms and economic capital in terms of credit riskmodeling. Both regulatory and economic capital are supposed to cover unexpected lossesresulting from banks’ lending operations to clients exposing different levels of default risk.Whereas holding regulatory capital is compulsory as a part of adherence to prudential regulations,holding economic capital is the banks’ own choice, on condition that its level reaches at least thelevel of regulatory capital. Worth mentioning, however, is the regulatory tendency to come closerto credit risk modeling and to allow banks to develop their own models for determining theamount of regulatory capital to hold. These models will most probably adopt and synthesize manyfeatures of the credit risk models already in use. This is one reason why comparing regulatory andeconomic capital today is becoming an insightful exercise for the regulatory decisions of thefuture.

2. Literature Review

In June 1999, the Basel Committee on Banking Supervision released a proposal to replace the1988 Accord with a more risk-sensitive framework. A concrete proposal in the form of aconsultative document, “The New Basel Capital Accord”, was presented in January 2001. Thisdocument proposed new regulatory rules for banks’ capital adequacy evaluations. The maininnovations related to credit and operational risk. In terms of credit risk, the New Basel CapitalAccord revised the old 1988 Basel Accord by proposing a more risk-sensitive methodology forassessing the default risk of banks’ clients. The risk inputs entering the final capital adequacycomputations were closely related to the risk characteristics of individual bank clients. In thissense, the proposed methodology opted for the adoption of ratings (developed by externalagencies or by banks themselves) in quantifying and signaling to the bank the default risk ofindividual borrowers. In a simpler version of the methodology (the standardized approach),ratings are directly associated with risk weights (for example, an A-rated asset would be assigneda risk weight of 50%, a BBB-rated asset would be assigned a risk weight of 100%, and so on). Ina more advanced approach (the Internal Rating Based, or IRB, approach), ratings represent thebasis for computing the probability of an obligor’s default. Default probabilities and other riskcharacteristics (loss given default, exposure at default) enter more complicated formulas fordetermining the risk weights of individual assets in regulatory capital estimations.


The consultative document was open to comments and feedback from the banking industry untilMay 2001. Subsequently, the Basel Committee on Banking Supervision updated the originaldocuments twice (in June and December 2001) and conducted three quantitative impact studies.The Committee intends to incorporate the feedback received from the banking industry and tocome up with the final version of the NBCA by the end of 2003. The active implementation of thedocument by internationally operating banks is supposed to start after the end of 2006. Thediscussion with banks concerning potential changes to the original NBCA (January 2001) hasrevolved around several topics:

• the construction of a new risk-weight curve in the IRB approach,• narrowing the gap between the capital estimations produced by the two versions of the IRB

approach (foundation and advanced),• adequate treatment of small and medium-sized enterprises,• the business cycle dependence of the IRB approach,• the necessity of implementing conservative credit-risk stress testing under the IRB approach.

In the banking industry, credit risk modeling has also been explored and extended since therelease of the four major credit risk models at the end of the last decade2. In this paper weconsider only two such models, CreditMetrics and CreditRisk+, which utilize, respectively, thestructural and the reduced-form approach to modeling default risk (see Duffie and Singleton,1998). In the structural approach, it is assumed that default is triggered when an unobservedvariable (obligor’s firm asset value) falls below a certain threshold level (firm’s outstanding debt).CreditMetrics extends this reasoning to rating downgrades by defining rating class-specificthreshold levels that mark the switch from one rating class to another in the event that the firm’sstandardized asset returns cross these threshold values. In the reduced-form literature, default ismodeled as an autonomous stochastic process that is not driven by any variable linked to theobligor’s firm capital structure or asset value. Particular formulations for the default process wereconsidered in Jarrow and Turnbull (1995) (exponential distribution), Jarrow, Lando and Turnbull(1997) (a continuous Markov chain) and Duffie and Singleton (1998) (a stochastic hazard rateprocess). CreditRisk+ represents the reduced-form approach by assuming that the average numberof defaults in each homogeneous class of obligors follows a Poisson distribution. The unifyingelement of the CreditMetrics and CreditRisk+ models is the Value at Risk (VAR) methodologyused in quantifying and provisioning for credit risk at the portfolio level. Even thoughCreditMetrics derives the portfolio value distribution and CreditRisk+ the portfolio lossdistribution at the end of the risk horizon, both models estimate economic capital such thatunexpected losses are covered by the estimated economic capital within an acceptable confidencelevel.

The KMV model (see, for instance, Crosbie, 1999) represents another step towards market-basedderivation of economic capital. Similarly to CreditMetrics, it uses the obligor’s equity pricestatistics to derive the value distribution of a given loan. Correlations are obtained automaticallyfrom the risk factors that determine the obligor firm values (equities). However, this methodrequires the assumption of complete markets, the validity of risk-neutral asset valuation and

2 We refer to JP Morgan’s Credit Metrics/Credit Manager model, Credit Suisse Financial Products’ CreditRisk+,KMV Corporation’s KMV model, and McKinsey’s CreditPortfolioView.


tradability of both the obligors’ equities and their debt in the bank portfolio. The KMV teamoffers unspecified remedies in cases where one of these preconditions is not satisfied (Sellers andDavidson, 1998), but open sources of credit risk literature offer no general solution of theseproblems.

This paper proposes a way around the said difficulties in the KMV approach by resorting to theso-called pricing-kernel methods of asset pricing (comprehensive expositions can be found in, forinstance, Campbell et al., 1997, and Cochrane, 2001). Asset tradability and market completenessare no longer necessary, and there are numerous possibilities for modeling default events thatdepend on systemic and idiosyncratic risk factors. Numerical approaches to calculating pricing-kernel-based asset values have also been developed in recent years (see, for instance, Ait-Sahaliaand Lo, 2000, or Rosenberg and Engle, 2002).

3. Methodology

The paper looks at Czech banks and their commercial loan market from the potential capitalrequirements point of view. Three pillars make up the main structure of our analysis. First, thetested bank loan portfolio is constructed in such a way as to reflect with some degree of realismthe rating distribution of a pool of Czech bank clients. Second, we take into account the randomnature of interest rates and other economic fundamentals that enter the loan valuation. Amongother things, this means that identifiable uncertainty factors in the loan characteristics which areusually treated in the market risk context (interest rates and exchange rates) were an integral partof the capital calculations as far as each of the tested approaches allowed. Third, when conductingmodel-based economic capital calculations, we follow the market loan pricing point of viewwherever possible (i.e. when the corresponding model allows it either explicitly or implicitly).This is done because we want to identify those elements of capital requirements which may beseen differently from the credit risk modeling and regulatory perspectives.

Our analysis utilizes a hypothetical portfolio containing 30 loans. This simplified portfolio mirrorsthe rating structure of a real loan portfolio obtained on the basis of a pool of corporate customersof six Czech banks3. Since ratings are the key input in many credit risk approaches, a simplifiedversion of Moody’s rating methodology for private firms has been applied to obtain ratings in ourreal sample of bank clients. Estimates of other inputs required by credit risk modeling which werenot available in the real bank data set were obtained using aggregate data from CNB databases.

We examined and compared the predictions of the NBCA with those delivered by theCreditMetrics and CreditRisk+ models. Following the last available consultative version of theNBCA guidelines (January 2001) we found that in our particular example the standardizedapproach of the NBCA predicted approximately the same level of capital as the credit risk modelsat the 95% confidence level. At the 99% confidence level, the internal credit risk models predicteda higher level of economic capital than the NBCA’s standardized approach, but these estimateswere still lower than the estimates of the NBCA’s IRB approach. We obtained different resultswhen applying the NBCA guidelines as formulated by the third quantitative impact survey, QIS 3(October 2002). Here, the results of both NBCA approaches (standardized and IRB) were more 3 We would like to thank Alena Buchtíková for making this data set available for our research purposes.


similar to each other, with the IRB requirement being slightly lower than the requirement of thestandardized approach. The results of both regulatory approaches were even lower than the levelof capital required by the various credit risk models.

We extend the analysis of economic capital by allowing both the bank lending rates and theforward zero rates used as discount factors in asset valuations to become random variables (a formof stress-testing). Floating lending rates and random changes in the forward zero curves were allimplemented using Monte Carlo simulations in the context of the CreditMetrics model. Asexpected, more uncertainty associated with the evolution of these variables required moreeconomic capital to be held by banks. However, the proposed changes in forward zero curves didnot impose significantly different levels of credit risk-related economic capital as compared withthe case of stable forward zero curves (but maintaining floating interest rates in both cases).Downward movements in the forward zero curves (translation or rotation) required higher levelsof capital at all confidence levels. This is an inconvenient consequence of the existing creditmodels (CreditMetrics and KMV in particular), which we strive to overcome by proposing amodel of our own.

The paper is structured as follows. In Section 4 we briefly describe the main characteristics of thereal bank and test portfolios and their estimation. We also mention the reasons why the modelscould not be implemented entirely on the basis of real Czech bank data. Section 5 presents the twoapproaches of the NBCA (standardized and IRB) and their estimates for credit risk-relatedregulatory capital. In Section 6 we outline the methodologies proposed by two widespread creditrisk models (CreditMetrics and CreditRisk+) and present their economic capital estimations. InSection 7 the assumption of fixed lending and forward zero rates is relaxed and consequently newestimates of economic capital are obtained. Section 8 outlines our own model of risky debt, itsvaluation and the resulting economic capital requirements, going along the structural lines of theoriginal KMV. Section 9 concludes. Finally, the Appendix outlines the technical details of theartificial loan portfolio generation, the RiskCalc model embodying Moody’s methodology forrating private firms, and an estimation procedure outline for our debt pricing model.

4. Description of the Test Portfolio

Our bank data set contains the balance sheets and profit and loss accounts of non-financial firmsthat were granted bank loans between 1994 and 2000. The CZ-NACE classification4, legal formand CNB loan classification5 (from 1997) were also available for each bank customer. Six Czechbanks provided the data to the CNB from 1994 until 1999, of which two banks terminatedcooperation in 2000. The banks reported only a fraction of their corporate portfolios. The exactselection procedure used by banks in choosing certain firms is not known. Also unknown is theproportion of reported versus unreported clients satisfying certain criteria. In this sense, weobserved a certain bias of the data providers towards non-reporting of loans in the last twocategories (4 and 5) but were not able to assess the direction and magnitude of this sampling biasin our results.

4 CZ-NACE (Czech abbreviation: OKEČ) represents the industry classification of economic activity in theCzech Republic.5 The CNB’s loan classification ranges from 1 to 5, with category 1 meaning standard, 2 watch, 3 nonstandard, 4doubtful and 5 loss loans.


Since our main goal was to assign ratings to banks’ corporate clients, we primarily focused ontheir default behavior. Default was defined as a credit event in which the loan classification of acertain company migrated from the 1st or 2nd category to any of the 3rd, 4th or 5th categories overthe considered risk horizon. Due to the short time length of our data set we had to focus on annualdefault rates. The largest number of defaults occurring over a one-year period was recordedbetween 1997 and 1998, representing 8% of all firms in the sample6. The sample-based annualdefault rates were 0.07% between 1998 and 1999 and 0% between 1999 and 2000. These lowdefault rates may be partially explained by the Czech economic recovery and by more prudentbank lending behavior during 1999–2000. Nevertheless, we think that the main reason isinsufficient default reporting by banks. Therefore, we preferred to restrict the reference data setonly to the accounting information collected in 1997 and the default events observed in 1998,assuming that default reporting by banks in that period was closer to the reality. While analyzing alonger time period would have been highly valuable, we considered that the sample-basedinformation over 1998–2000 painted a biased picture about corporate default and, consequently, itwas not used in modeling the rating structure of the test portfolio.

The annual default rate of 8% in the reference data set was significantly higher than the averagevalue of 1.5% usually used by Moody’s in the context of Western European economies. However,volume-based information about the loan defaults of individuals and corporates in the entirebanking sector revealed an annual default rate of approximately 20%. We considered that neither1.5% nor 20% would be the appropriate annual default rate for our artificial portfolio7 and, ingeneral, for a typical Czech bank portfolio of corporate loans. Without any other more reliablesource of information, we used the 8% default rate as indicated by our sample bank data tocalibrate the probit model (see Subsection A1.1 of the Appendix).

Table 1: S&P’s Rating Class-Specific One-Year Default Rates

Rating One-year default rate (%) Cut-off values for definingratings in the bank portfolio (%)

AAA 0 -AA 0 -A 0.06 0.03

BBB 0.18 0.12BB 1.06 0.62B 5.2 3.13

CCC 19.79 12.495

Source: Credit Metrics – Technical Document.

To assign ratings to each firm we used the calculated default rates (Appendix A1.2) and the tablescontaining cumulative default rates published by different rating agencies. Even though Moody’srating methodology was used, we preferred to calibrate our results to Standard & Poor’s (S&P) 6 The reference sample included only bank clients that were present both in 1997 and 1998 (663 firms). Toexamine only one-year default behavior, those enterprises which were already in default in 1997 were alsoeliminated. In the end we obtained a data set containing 606 firms.7 These figures reflected assumptions that were not applicable in our case. For example, the 1.5% level wasbased on the Western experience, while the 20% level was volume-based and represented both firms’ andindividuals’ default behavior.


ratings. This was done since (a) the New Basel Capital Accord assigned risk weights based onS&P ratings and (b) the inputs in the credit risk models were, to a great extent, based on S&Pdata. For calibration purposes we used a 1996 S&P cumulative one-year default rate matrix thatlooked like that in Table 1.

The probabilities given in the second column are rating class-specific default probabilitiespublished by S&P. The third column contains the probabilities that mark the transition from onerating class to another in our model. They are the midpoints in the intervals determined by theone-year default probabilities given in the second column. For example, if the estimated defaultprobability of a certain firm belonged to the interval [0, 0.03), an AA rating grade was assigned tothat firm. If the estimated default probability belonged to the interval [0.03, 0.12), then an A gradewas assigned and so on. Based on this mapping procedure, each firm that was present in the 2000data set was marked with a certain rating grade. The resulting rating structure and the loanclassification of the pooled bank portfolio for 2000 are presented in Table 2.

Table 2: The Pooled Bank Portfolio Structure in 2000 According to Loan Classification andRatings (Number of Firms/Percentage)

AA A BBB BB B CCC Total1 15/1.45 5/0.48 26/2.51 87/8.41 580/56.04 89/8.60 802/77.492 1/0.10 0 1/0.10 19/1.84 150/14.49 23/2.22 194/18.743 0 0 0 0 12/1.16 11/1.06 23/2.224 0 0 0 0 3/0.29 3/0.29 6/0.585 0 0 0 0 1/0.10 90.87 10/0.97

Total 16/1.55 5/0.48 27/2.61 106/10.24 746/72.08 135/13.04 1035/100Source: Own computation.

Next, we constructed an artificial portfolio incorporating as much real information as possible.Outside the ratings, the bank data set did not contain information regarding loan volumes ormaturities, charged interest rates and borrower asset returns or recovery rates. Since theseparameters represent required inputs into many regulatory and internal credit risk models, weconstructed proxy variables based on data available at the macro level or obtained them as randomdrawings from known distributions. In what follows we describe the manner in which these inputswere generated. The main information source was the SUD database of the Czech National Banksupervisory body, which contains yearly information about the residual maturity of Czech bankloans, their category and the borrower CZ-NACE category. SUD also categorizes loans accordingto the charged interest rate.

Ratings and Exposures

We adjusted the rating structure displayed in Table 2 to reflect the following changes:• All bank clients in loan category 5 were eliminated. Loans in this category are loss loans,

which are usually covered by provisions created in the current period. Moreover, the 5th

category is an absorbing state. Once a loan falls into this category, there is a high probabilitythat it will not recover. These loans pose a vacuous problem from the risk managementperspective, since the future state of these loans is associated with almost no uncertainty.


• The 8.6% of firms with a CCC rating were removed from category 1 and added to categories3 and 4. We assumed that the 8.6% outcome reflected the imperfections of our model. Czechbanks monitor the creditworthiness of their clients, thus loans falling in the first category areunlikely to be granted a CCC grade.

• The rating structure was adjusted to resemble the loan volume configuration at the end ofDecember 2000 as closely as possible (as shown in Table 3).

Table 3: Loan Volumes by Category Granted by Czech Banks, as of End 2000

Category Volume (CZK bn) Proportion (%)1 607.235 68.582 85.811 9.693 54.577 6.164 26.982 3.055 110.834 12.52

Source: CNB.

After all these changes, the rating structure of our artificial (test) portfolio took the form shown inTable 4. To have a fair representation of all ratings in each loan category, one needs a minimumof 30 assets in the artificial portfolio.

Table 4: Rating Structure of the Artificial Portfolio – Number of Assets in Each Loan Categoryand Rating Classes

Loancategory AA A BBB BB B CCC Total

1 1 1 2 3 14 0 212 0 0 0 1 2 1 43 0 0 0 0 1 2 34 0 0 0 0 0 2 2

Total 1 1 2 4 17 5 30

In our test portfolio the exposure of an asset in a certain loan category represents the ratio betweenthe total loan volume and the number of assets in that category. For example, all assets belongingto category 1 (21 in the test portfolio) have an exposure of CZK 607.235/21=28.915 billion.

Maturities

Loans with a maturity exceeding five years are sparsely represented in the Czech bank portfolios.For this reason, we considered maturities that ranged from one to five years only. Maturity wasassigned to individual assets by drawing random numbers from the interval [1,5] according to theuniform distribution and then rounding these numbers to the nearest integer.

Interest Rates

We computed the mean and standard deviation of the lending rates for each loan category (usingthe SUD data set). To assign interest rates to assets in our portfolio, we randomly drew numbersfrom normal distributions described by the estimated means. Standard deviations were in generalreduced to prevent interest rates from deviating too much from these mean values.


Asset Return Correlations

We grouped firms in the bank data set according to ratings and loan classification and found theCZ-NACE category that was the most frequently represented in each group. For example, in thegroup of firms with rating AA and loan classification 1 the largest number of firms belonged toCZ-NACE 51. If in a certain group no dominating CZ-NACE could be found, we selectedrandomly the representative figure from those that were represented in that group. The resultingCZ-NACE structure was mapped to the test portfolio. Having assigned a CZ-NACE label to eachasset in the test portfolio, we used the price index characteristic of the corresponding branch as aproxy variable for that asset’s returns8. Asset return correlations were determined by computingcorrelations among price indices.

Finally, having generated collateral and recovery rates (Appendix A1.3) we obtained a portfolioof 30 loans, whose characteristics are displayed in Table 5.

Table 5: Portfolio Composition and Individual Assets’ Characteristics

Asset CZ-NACE

Regulatoryloan class

Rating Loanvolume

Maturity Interest rate(%)

Type ofcollateral

Recoveryrate (%)

1 51 1 AA 28.916 3 6.5 6 02 36 1 A 28.916 1 7.3 6 03 74 1 BBB 28.916 3 7.5 6 04 31 1 BBB 28.916 5 7.6 6 05 74 1 BB 28.916 5 8.2 4 71.436 20 1 BB 28.916 5 8.5 1 94.347 51 1 BB 28.916 1 8.7 6 08 28 1 B 28.916 3 8.8 2 1009 15 1 B 28.916 4 9.5 6 0

10 51 1 B 28.916 2 10.5 1 94.3411 52 1 B 28.916 2 11.0 5 5012 29 1 B 28.916 1 11.1 6 013 70 1 B 28.916 1 11.3 5 5014 74 1 B 28.916 2 11.6 4 71.4315 50 1 B 28.916 2 11.7 1 94.3416 24 1 B 28.916 1 11.9 5 50.0017 45 1 B 28.916 2 12.1 1 94.3418 60 1 B 28.916 2 12.2 1 94.3419 40 1 B 28.916 3 12.5 3 89.2920 25 1 B 28.916 2 12.6 1 94.3421 65 1 B 28.916 4 13.0 4 71.4322 51 2 BB 21.452 2 8.4 1 94.3423 25 2 B 21.452 5 10.4 1 94.3424 29 2 B 21.452 3 11.5 6 025 55 2 CCC 21.452 3 13.5 5 5026 45 3 B 18.192 2 11.9 1 94.3427 28 3 CCC 18.192 5 12.4 1 94.3428 21 3 CCC 18.192 4 12.2 1 94.3429 21 4 CCC 13.491 5 14.3 4 71.4330 37 4 CCC 13.491 2 15.4 1 94.34

Source: Own computation.

8At the outset, all price indices were deflated by the PPI in order to eliminate the systemic inflationary influencein their evolution.


5. The BIS Approach to Economic Capital Determination

At the time when the present research was conducted, the latest officially released consultativedocument describing the New Basel Capital Accord approach was the one from January 2001.Since the release of this version, much further discussion has been going on and the Committeehas proposed various modifications. The issuance of another consultative version is expected inthe second quarter of 2003. At this moment the last available version of this document is in theform of technical guidance to the third quantitative survey (QIS 3). This section compares both ofthe above-mentioned versions of the Basel document.

5.1 The Standardized Approach According to the January 2001 Consultative Document

When computing the capital requirements for credit risk according to the standardized approach,we applied the comprehensive treatment of collateral with the standard supervisory haircuts. Thecomprehensive approach according to the January 2001 NBCA Consultative Document meansthat banks apply haircuts to the market value of collateral in order to protect against pricevolatility. A weight (floor factor) is applied to the collateralized portion of the exposure afteradjusting for the haircut.

All claims in the simulated portfolio were assigned to the group of exposures “Claims oncorporates” and were given the risk weights corresponding to their rating (Table 6).

Table 6: Risk Weights for the Group of Exposures “Claims on Corporates”

AAA to AA- A+ to A- BBB+ to BB- Below BB- Unrated

Risk weights 20% 50% 100% 150% 100%Source: Basel Committee on Banking Supervision (January 2001).

When setting the risk weights and risk-weighted assets, the quality and the amount of collateralwere also considered. The NBCA defines the eligible collateral and specifies the appropriatehaircuts. The eligible collateral consists of bank deposits with the lending bank, BB- or above-rated securities issued by sovereigns and public sector entities, BBB- or above-rated bank andcorporate securities, equities included in a main index and gold. The haircuts are used to computethe adjusted value of the collateral according to the formula

CA = C / (1 + HV + HC + HFX),

where HV, HC and HFX are, respectively, the exposure, the collateral and the currency haircut.


The value of risk-weighted assets with adjusted collateral is computed according to the followingequation:

RWA = RW x (V – (1-w) x CA),

where• RW is the risk weight of the exposure (of the obligor)• V is the exposure amount• CA is the adjusted value of collateral• w is a floor factor (in this case 0.15).

When the assets are covered by bank guarantee, the value of risk-weighted assets is given by:

RWA = V x (w x r + (1-w) x g),

where• r is the risk weight of the obligor• g is the risk weight of the guarantor.

For other assets not covered by acceptable collateral or bank guarantee, the risk-weighted value isRWA = V x RW.

The capital requirement is set as 8 per cent of the sum of all risk-weighted assets in the portfolio:

��

iiRWA0.08CR =�

iiRWAx (0.08 ) .

In our case the type of collateral offered was taken into account in only two cases (Assets 8 and19) and influenced the risk-weighted assets through the adjusted value of the collateral. In the caseof those assets which were covered by bank guarantee (Assets 6, 10, 15, 17, 18, 20, 22, 23, 26–28and 30), the calculation took into account the risk weight of the guarantor (assuming g = 0.2). Forthe remaining assets, the collateral offered did not belong to the accepted collateral and the claimskept the risk weights given in Table 9.

Table 7 presents the partial results for the individual claims in the portfolio. The total capitalrequirement using this method is CZK 51.8 billion.


Table 7: Capital Requirement Computation – Standardized Approach (January 2001)

Asset Rating Volume(CZK billion)

Risk Weights (RW)without guarantee

influence

Risk-WeightedAssets (RWA)with collateraland guarantee

influence

Adjusted value ofcollateral

(CA)

Capitalrequirement

1 AA 28.92 0.20 5.78 0 0.462 A 28.92 0.50 14.46 0 1.163 BBB 28.92 1.00 28.92 0 2.314 BBB 28.92 1.00 28.92 0 2.315 BB 28.92 1.00 28.92 0 2.316 BB 28.92 1.00 9.25 influence on RWA 0.747 BB 28.92 1.00 28.92 0 2.318 B 28.92 1.50 6.51 28.92 0.529 B 28.92 1.50 43.37 0 3.47

10 B 28.92 1.50 11.42 influence on RWA 0.9111 B 28.92 1.50 43.37 0 3.4712 B 28.92 1.50 43.37 0 3.4713 B 28.92 1.50 43.37 0 3.4714 B 28.92 1.50 43.37 0 3.4715 B 28.92 1.50 11.42 influence on RWA 0.9116 B 28.92 1.50 43.37 0 3.4717 B 28.92 1.50 11.42 influence on RWA 0.9118 B 28.92 1.50 11.42 influence on RWA 0.9119 B 28.92 1.50 10.46 25.82 0.8420 B 28.92 1.50 11.42 influence on RWA 0.9121 B 28.92 1.50 43.37 0 3.4722 BB 21.45 1.00 6.86 influence on RWA 0.5523 B 21.45 1.50 8.47 influence on RWA 0.6824 B 20.38 1.50 30.57 0 2.4525 CCC 21.45 1.50 32.18 0 2.5726 B 18.19 1.50 7.19 influence on RWA 0.5827 CCC 18.19 1.50 7.19 influence on RWA 0.5828 CCC 18.19 1.50 7.19 influence on RWA 0.5829 CCC 13.49 1.50 20.24 0 1.6230 CCC 13.49 1.50 5.33 influence on RWA 0.43

51.84

Source: Own computation.Note: For the purposes of computing capital adequacy under the NBCA approaches, the volumes of

assets were lowered by the amount of specific provisions which would be created following theCNB regulatory norms (this only happened in the case of Asset 24, for which the amount ofspecific provisions would be approximately CZK 1.07 billion).

5.2 The Internal Rating Based (IRB) Approach According to the January 2001Consultative Document

When applying the Internal Ratings Based (IRB) approach to compute the capital requirements,we followed the foundation approach. This means that the bank may use its own estimates for thedefault probabilities of individual obligors. However, the assigned PD may not be lower than0.03%. The other risk components (LGD and EAD) are set according to the NBCA guidelines.


In our portfolio, the default probabilities of the individual assets were taken from the S&P matrix.Only in the case of one asset (Asset 1) was the probability of default lower than the minimumNBCA level; therefore, we had to change it to the minimum accepted value of 0.03%.

Loss Given Default (LGD) was set by taking into account the collateral of the individual claims:• Claims secured by collateral type 1 (bank guarantee) were assigned LGD = 0.5. In the case of

bank guarantees, the risk weight (RW) of the guarantor is taken into account when calculatingRWA.

• Claims fully secured by collateral types 2 and 3 (cash and securities respectively) LGD* = wx LGD, where LGD = 0.5.

• Claims secured by collateral type 4 (commercial real estate) were assigned LGD = [1-(0.2 x(C/V)/1.4) ] x 0.5

• Claims secured by collateral type 5 (other) or type 6 (unsecured) were assigned LGD = 0.5.

We will use the following notation:• PD – probability of default• M – maturity• b – sensitivity of the maturity adjustment factor to M• N(x) – cumulative normal distribution function• G(z) – inverse of the cumulative normal distribution function.

Further, BRW is defined as the benchmark risk weight for corporate exposures. It is a function theexposure’s PD:

BRW (PD) = 976.5 � N (1.118 � G(PD) + 1.288) � (1 + .0470 � (1 - PD)/PD0.44 ).

The baseline risk weight formula is

RW = min {(LGD/50) x BRW (PD), 12.5 x LGD}.

In cases where the claim maturity is explicitly known, it is necessary to include this piece ofinformation in the risk weight calculation. This is done according to the following formula:

RW = min {(LGD/50) x BRW (PD) x [1 + b (PD) x (M - 3)], 12.5 x LGD}.

In our simulated portfolio the maturity of all claims was known. However, because at the time ofcomputation the NBCA had no specifications regarding coefficient “b”, we had to apply the firstrule for setting RW.


The capital requirement is determined, in the same way as in the standardized approach, as 8 percent of the risk-adjusted capital: ��

iiRWA0.08CR .

Table 8 presents the partial results for the individual claims of the simulated portfolio. The totalcapital requirement that we obtain using this approach is CZK 165.46 billion.

Table 8: Capital Requirement Computation – IRB Approach (January 2001)


PD(S&P)

PD(NBCA) BRW LGD

(NBCA)

(LGD/50)x

BRW(PD)

12.5 xLGD RWA

Capitalrequire

ment1 AA 28.92 0.0000 0.0003 14.07 0.50 0.14 6.25 4.07 0.332 A 28.92 0.0006 0.0006 21.37 0.50 0.21 6.25 6.18 0.493 BBB 28.92 0.0018 0.0018 42.21 0.50 0.42 6.25 12.21 0.984 BBB 28.92 0.0018 0.0018 42.21 0.50 0.42 6.25 12.21 0.985 BB 28.92 0.0106 0.0106 129.67 0.43 1.11 5.36 32.14 2.576 BB 28.92 0.0106 0.0106 129.67 0.50 1.30 6.25 37.50 3.007 BB 28.92 0.0106 0.0106 129.67 0.50 1.30 6.25 37.50 3.008 B 28.92 0.0520 0.0520 338.83 0.08 0.51 0.94 14.70 1.189 B 28.92 0.0520 0.0520 338.83 0.50 3.39 6.25 97.98 7.84

10 B 28.92 0.0520 0.0520 338.83 0.50 3.39 6.25 97.98 7.8411 B 28.92 0.0520 0.0520 338.83 0.50 3.39 6.25 97.98 7.8412 B 28.92 0.0520 0.0520 338.83 0.50 3.39 6.25 97.98 7.8413 B 28.92 0.0520 0.0520 338.83 0.50 3.39 6.25 97.98 7.8414 B 28.92 0.0520 0.0520 338.83 0.43 2.90 5.36 83.98 6.7215 B 28.92 0.0520 0.0520 338.83 0.50 3.39 6.25 97.98 7.8416 B 28.92 0.0520 0.0520 338.83 0.50 3.39 6.25 97.98 7.8417 B 28.92 0.0520 0.0520 338.83 0.50 3.39 6.25 97.98 7.8418 B 28.92 0.0520 0.0520 338.83 0.50 3.39 6.25 97.98 7.8419 B 28.92 0.0520 0.0520 338.83 0.08 0.51 0.94 14.70 1.1820 B 28.92 0.0520 0.0520 338.83 0.50 3.39 6.25 97.98 7.8421 B 28.92 0.0520 0.0520 338.83 0.43 2.90 5.36 83.98 6.7222 BB 21.45 0.0106 0.0106 129.67 0.50 1.30 6.25 27.82 2.2323 B 21.45 0.0520 0.0520 338.83 0.50 3.39 6.25 72.69 5.8224 B 20.38 0.0520 0.0520 338.83 0.50 3.39 6.25 69.05 5.5225 CCC 21.45 0.1979 0.1979 665.20 0.50 6.65 6.25 134.08 10.7326 B 18.19 0.0520 0.0520 338.83 0.50 3.39 6.25 61.64 4.9327 CCC 18.19 0.1979 0.1979 665.20 0.50 6.65 6.25 113.70 9.1028 CCC 18.19 0.1979 0.1979 665.20 0.50 6.65 6.25 113.70 9.1029 CCC 13.49 0.1979 0.1979 665.20 0.43 5.70 5.36 72.27 5.7830 CCC 13.49 0.1979 0.1979 665.20 0.50 6.65 6.25 84.32 6.75

165.46

Source: Own computation.Note: For the purposes of computing capital adequacy under the NBCA (January 2001) approach, the

volumes of assets were lowered by the amount of specific provisions which would be createdfollowing the CNB regulatory norms (this only happened in the case of Asset 24, for which theamount of specific provisions would be approximately CZK 1.07 billion).


5.3 The Standardized Approach According to Technical Guidance to QIS 3, October2002

All claims in the simulated portfolio were assigned to the group of exposures “Claims oncorporates” and were given the risk weights corresponding to their rating grade. At this stage boththe rating groups and the risk weights assigned to them remained the same as in the consultativedocument from January 2001 (see Subsection 5.1).

Calculating the capital requirement, we followed the comprehensive approach. Thecomprehensive approach here means that banks calculate their adjusted exposure to a counterpartyfor capital adequacy purposes by taking into account the effects of that collateral. Banks adjustboth the amount of the exposure to the counterparty and the value of any collateral received totake account of future fluctuations in the value of each. (Basel Committee on BankingSupervision, QIS3, October 2002.)

Accordingly, the amount of exposure after risk mitigation for a collateralized loan is given by

E* = max { 0, [E x (1 + HV) – C x (1 – HC – HFX)]},

Where the haircut coefficients are defined as before and• E* is the exposure amount after risk mitigation,• E is current value of the exposure,• C is the current value of the collateral received.

The exposure amount was adjusted (due to the quality and amount of the collateral) in the case ofAssets 8 and 19 only, because only these two were covered by eligible collateral. In the othercases, the collateral was either not considered or the quality of the collateral (bank guarantee) wastaken into account when assigning the risk weights. In the case of bank guarantees the protectedportion was assigned the risk weight of the protection provider (assuming RW = 0.2). This wasthe case for Assets 6, 10, 15, 17, 18, 20, 22, 23, 26–28 and 30.

For each claim in the portfolio, the risk-weighted asset value can be computed as the product ofthe risk weight of the exposure and the exposure amount (after risk mitigation if applicable):

RWA = RW x V.

The capital requirement is taken equal to 8 per cent of the risk-weighted asset sum, as is usual inthe standardized and IRB approaches: ��

iiRWA0.08CR .

Table 9 presents the partial results for the individual claims in the portfolio. The total capitalrequirement using this method is CZK 46.9 billion.


Table 9: Capital Requirement Computation – Standardized Approach (October 2002)


Risk Weights (RW)without guarantee

influence

Exposureamount after

risk mitigation(E*)

Risk-Weighted Assets(RWA) with collateral

and guaranteeinfluence

Capitalrequirement

1 AA 28.92 0.20 28.92 5.78 0.462 A 28.92 0.50 28.92 14.46 1.163 BBB 28.92 1.00 28.92 28.92 2.314 BBB 28.92 1.00 28.92 28.92 2.315 BB 28.92 1.00 28.92 28.92 2.316 BB 28.92 1.00 guarantee 5.78 0.467 BB 28.92 1.00 28.92 28.92 2.318 B 28.92 1.50 0 0.00 0.009 B 28.92 1.50 28.92 43.37 3.47

10 B 28.92 1.50 guarantee 5.78 0.4611 B 28.92 1.50 28.92 43.37 3.4712 B 28.92 1.50 28.92 43.37 3.4713 B 28.92 1.50 28.92 43.37 3.4714 B 28.92 1.50 28.92 43.37 3.4715 B 28.92 1.50 guarantee 5.78 0.4616 B 28.92 1.50 28.92 43.37 3.4717 B 28.92 1.50 guarantee 5.78 0.4618 B 28.92 1.50 guarantee 5.78 0.4619 B 28.92 1.50 3.47 5.20 0.4220 B 28.92 1.50 guarantee 5.78 0.4621 B 28.92 1.50 28.92 43.37 3.4722 BB 21.45 1.00 guarantee 4.29 0.3423 B 21.45 1.50 guarantee 4.29 0.3424 B 20.38 1.50 21.45 32.18 2.5725 CCC 21.45 1.50 21.45 32.18 2.5726 B 18.19 1.50 guarantee 3.64 0.2927 CCC 18.19 1.50 guarantee 3.64 0.2928 CCC 18.19 1.50 guarantee 3.64 0.2929 CCC 13.49 1.50 13.49 20.24 1.6230 CCC 13.49 1.50 guarantee 2.70 0.22

46.90


5.4 The Internal Rating Based (IRB) Approach According to Technical Guidance to QIS3 (from October 2002)

When applying the Internal Ratings Based (IRB) approach to the capital requirementcomputations, we followed the same approach as in Subsection 5.2 – the foundation approach.That is, the bank may use its own estimates for the default probabilities of individual obligors; theother risk components (LGD and EAD) are set according to the NBCA guidelines.

As in the case mentioned above, the default probabilities of the individual assets were taken fromthe S&P matrix. Only in the case of one asset (Asset 1) was the probability of default lower thanthe minimum NBCA level; therefore, we had to change it to the minimum accepted level of0.03%.


However, the calculation of risk-weighted assets (followed in the case of all assets in oursimulated portfolio) is different from that used under the January 2001 Consultative Document:

Risk-weighted assets (RW) = K x 12.50 x EAD,

where

Capital requirement (K) = LGD x N [(1 – R)-0.5 x G (PD) + (R / (1 – R)) 0.5 x G (0.999)] x (1 –1.5 x b (PD)) -1 x (1 + (M – 2.5) x b (PD))

Maturity adjustment b(PD) = (0.08451 – 0.05898 x log (PD)) 2

Correlation (R) = 0.12 x (1 – EXP (-50 x PD)) / (1 – EXP (-50)) + 0.24 x [1 – (1 - EXP (-50 xPD)) / (1 – EXP (-50))].

Loss Given Default was set by taking into account the collateral of the individual claims:• Claims secured by collateral type 1 (bank guarantee) were assigned LGD = 0.45. In the case

of bank guarantees, the risk characteristics (PD and LGD) of the guarantor are taken intoaccount when calculating RWA.

• Claims fully secured by collateral types 2 and 3 (cash and securities respectively) LGD* =Max {0, LGD x [(E* / E)]}, where LGD = 0.45.

• Claims secured by collateral type 4 (commercial real estate) were assigned LGD = 0.35• Claims secured by collateral type 5 (other) or type 6 (unsecured) were assigned LGD 0.45.

The capital requirement is determined as usual: ��

iiRWA0.08CR , which is equal to CZK

44.8 billion in our example. Table 10 presents the partial results for the individual claims in theportfolio.

Table 10: Capital Requirement Computation – IRB Approach (October 2002)


PD(S&P)

PD(NBCA)

LGD(QIS3)

Exposure amountafter risk

mitigation (E*)

E*/ V Correlation(R)

Maturityadjustment

(b)

Capitalrequirement

(K)

Risk-weightedassets(RW)

Capitalrequirement

1 AA 28.92 0 0.0003 0.45 28.92 1.00 0.24 0.09 0.01 2.24 0.182 A 28.92 0.0006 0.0006 0.45 28.92 1.00 0.24 0.08 0.01 3.82 0.313 BBB 28.92 0.0018 0.0018 0.45 28.92 1.00 0.23 0.06 0.02 8.39 0.674 BBB 28.92 0.0018 0.0018 0.45 28.92 1.00 0.23 0.06 0.02 8.39 0.675 BB 28.92 0.0106 0.0106 0.35 28.92 1.00 0.19 0.04 0.05 18.25 1.466 BB 28.92 0.0106 0.0106 0.45 guarantee x 0.24 0.08 0.01 3.82 0.317 BB 28.92 0.0106 0.0106 0.45 28.92 1.00 0.19 0.04 0.06 23.47 1.888 B 28.92 0.052 0.052 0 0 0.00 0.13 0.03 0.00 0.00 0.009 B 28.92 0.052 0.052 0.45 28.92 1.00 0.13 0.03 0.13 47.28 3.78

10 B 28.92 0.052 0.052 0.45 guarantee x 0.24 0.08 0.01 3.82 0.3111 B 28.92 0.052 0.052 0.45 28.92 1.00 0.13 0.03 0.13 47.28 3.7812 B 28.92 0.052 0.052 0.45 28.92 1.00 0.13 0.03 0.13 47.28 3.7813 B 28.92 0.052 0.052 0.45 28.92 1.00 0.13 0.03 0.13 47.28 3.7814 B 28.92 0.052 0.052 0.35 28.92 1.00 0.13 0.03 0.10 36.77 2.9415 B 28.92 0.052 0.052 0.45 guarantee x 0.24 0.08 0.01 3.82 0.3116 B 28.92 0.052 0.052 0.45 28.92 1.00 0.13 0.03 0.13 47.28 3.7817 B 28.92 0.052 0.052 0.45 guarantee x 0.24 0.08 0.01 3.82 0.3118 B 28.92 0.052 0.052 0.45 guarantee x 0.24 0.08 0.01 3.82 0.3119 B 28.92 0.052 0.052 0.054 3.47 0.12 0.13 0.03 0.02 5.67 0.4520 B 28.92 0.052 0.052 0.45 guarantee x 0.24 0.08 0.01 3.82 0.3121 B 28.92 0.052 0.052 0.35 28.92 1.00 0.13 0.03 0.10 36.77 2.9422 BB 21.45 0.0106 0.0106 0.45 guarantee x 0.24 0.08 0.01 2.83 0.2323 B 21.45 0.052 0.052 0.45 guarantee x 0.24 0.08 0.01 2.83 0.2324 B 20.38 0.052 0.052 0.45 21.45 1.00 0.13 0.03 0.13 35.08 2.8125 CCC 21.45 0.1979 0.1979 0.45 21.45 1.00 0.12 0.02 0.27 71.93 5.7526 B 18.19 0.052 0.052 0.45 guarantee x 0.24 0.08 0.01 2.40 0.1927 CCC 18.19 0.1979 0.1979 0.45 guarantee x 0.24 0.08 0.01 2.40 0.1928 CCC 18.19 0.1979 0.1979 0.45 guarantee x 0.24 0.08 0.01 2.40 0.1929 CCC 13.49 0.1979 0.1979 0.35 13.49 1.00 0.12 0.02 0.21 35.18 2.8130 CCC 13.49 0.1979 0.1979 0.45 guarantee x 0.24 0.08 0.01 1.78 0.14

44.79Source: Own computation.


6. Commercial Risk-Measurement Models

6.1 CreditMetrics

In the CreditMetrics model, risk is associated with changes in the portfolio value caused bychanges in the credit quality of individual obligors (downgrades or default) over the consideredrisk period (usually taken as one year). The risk analysis undertaken by CreditMetrics consists oftwo distinct parts. The analytical part derives primary risk measures such as means, variances andstandard deviations at the asset and portfolio level. Monte Carlo simulation generates a simulatedportfolio value distribution at the risk horizon. Based on this distribution, estimates of economiccapital can be obtained at different confidence levels. Because the aim of this paper is to estimateeconomic capital, we focus on the application of the Monte Carlo method9.

Monte Carlo Simulation

The Monte Carlo simulation is a numerical method that makes inferences about unknownparameters of a stochastic process based on a large number of random draws from thedistributions supposed to drive the given process. Intuitively, it consists in obtaining a large set ofpotential realizations of the real process and consequently in aggregating the statisticalinformation thus obtained to estimate the unknown parameters. In CreditMetrics, this is done toestimate the portfolio value distribution at the risk horizon. Because changes in asset and portfoliovalue are caused by changes in the credit quality of individual assets (upgrades and downgrades)the Monte Carlo simulation is a tool that randomly generates potential changes in the actual ratingstructure of the portfolio.

More precisely, the Monte Carlo simulation performs a large number of random draws from amultinomial normal distribution (“scenarios”, in CreditMetrics terminology). Each scenariocontains the same number of components (real numbers) as the number of assets in the portfolio.Each component of each scenario is compared with pre-defined, rating class-specific thresholdvalues marking the switch from one rating class to another. In this way, within each scenario, anew rating is assigned to each asset (obligor) in the portfolio.

Moreover, in case of non-default, each asset i is revalued according to the formula:

� � � �,

f1

Fr

f1

rV

1T

1tTg

T

iitg

t

igi

i

i

i

��

�

�

�

�

�

�

� (1)

where ri and Fi are the loan interest payments and the face value of the loan respectively, and ftg

are the annualized forward zero rates for the years 1 to Ti, applicable to the rating class g' (here Ti

is the maturity of the loan). In this specification it is assumed that the present rating changes fromg to g' over the one-year period. In the case of default the present value of the loan is computed asthe product of the face value of the loan and a recovery rate. 9 In general, we do not go into a detailed coverage of the methodologies proposed by these models. This wasdone in our previous CNB working paper (see Derviz and Kadlčáková, 2001) specially dedicated to the subject.Here, our aim is to derive estimations of economic capital according to different models.


Note that the way of defining the future loan value as a random variable implies that thedistribution of this variable would be taken with respect to the risk-neutral probability (RNP).CreditMetrics works with the assumption that such a probability is well-defined for the studiedeconomy (cf. a more general view of the subject in our own model, particularly Subsection 7.1).Under the risk-neutral probability, in contrast to the “physical” one, zero forward rates areunbiased estimates of the future spot interest rates. That is, the capital requirements underCreditMetrics are also derived from RNP. This might lead to certain discrepancies between theCreditMetrics interpretation by the market (which is based on RNP) and its interpretation by theregulator (based on the physical probability).

To obtain the portfolio value distribution and derive the economic capital requirement inaccordance with the CreditMetrics model, we conducted a Monte Carlo simulation with 10,000random draws. Each scenario contained 30 correlated random draws from the standard normaldistribution. Each element of each scenario represented a standardized return corresponding toone of the 30 assets belonging to the portfolio. Comparing the elements of the scenario with thethreshold values characteristic of each rating category, new ratings were assigned to each of the30 assets. Summing up the values of the 30 assets thus obtained, a new portfolio value resulted foreach scenario. Since eight potential new ratings were possible for each of the 30 assets at the endof the year, the total number of potential portfolio values was 830. In practice, this number was farlower, as some rating class migrations had a zero probability of realization. Figure 1 shows thedistribution of our portfolio value expected at the end of 2000 for the year 2001. On the horizontalaxis are the non-overlapping intervals within which the portfolio value falls, while on the verticalaxis are the frequencies with which these portfolio value realizations occurred within each intervalin our simulation.

Figure 1: Outcomes of the Monte Carlo Simulation for the Loan Portfolio Value

0

1000

2000

3000

4000

660 680 700 720 740 760 780 800 820 840 860 880

Series: VPSample 1 10000Observations 10000

Mean 845.7842Median 855.1216Maximum 870.9298Minimum 663.7789Std. Dev. 22.49250Skewness -1.830517Kurtosis 6.928587

Jarque-Bera 12015.40Probability 0.000000

Source: Own computation.Note: To the right are statistics that characterize the portfolio value distribution. In addition to basic

descriptive statistics, the table displays skewness, kurtosis and the Jarque-Bera test statisticvalue.

Portfolio value

Frequency


Economic capital is obtained as the difference between the mean of the portfolio value and a p-percentile (p is usually assumed to be 1%, 2% or 5%):

Economic Capital = Mean of the Portfolio Distribution – p-percentile.

In the case of a discrete portfolio distribution, the p-percentile is obtained by looking at the lowestportfolio value whose cumulative frequency exceeds p%. An interpretation of the p- percentile isthat in p% of cases we can expect the portfolio value to take values lower than the p-percentileover the one-year period. For example, in our case we can expect that only 100 times in 10,000cases could the portfolio value reach a value lower than CZK 767.89 billion (in other words weknow with 99% probability that the portfolio value will be higher than CZK 767.89 billion at theyear-end). To cover this high loss, however unlikely it is, the bank must keep economic capital ofCZK 77.88 billion.

The 1% and 5%-percentiles take the values CZK 767.89 billion and CZK 796.62 billionrespectively. Based on these estimations, the bank’s need for economic capital at differentconfidence levels is:

1%-percentile 5%-percentile Mean

99%

Economic capital

95%

Economic capital

767.89 796.62 845.78 77.88 49.16

6.2 CreditRisk+

CreditRisk+ is suitable for assessing credit risk in portfolios containing a large number of obligorswith small default probabilities. The model groups bank customers according to their commonexposure. The common exposure of an obligor represents the ratio between his bank exposure anda selected unit of exposure (CZK 1 billion in our case). Bank clients can be grouped inhomogeneous “bands” that contain obligors with the same common exposure.

For obligor i, we introduce the following notations, taken from the CreditRisk+ technical

document: Li – exposure, Pi – default probability, LLi

i �� – common exposure, �i – rounded

common exposure, iii P�� – expected loss. In addition, at the given Band j-level, �j is the

common exposure in units of L, ��

��

jBandiij – expected loss in Band j in units of L,

j

jj

�

�� –

expected number of defaults in Band j.

The risk assessment at the asset level consists in estimating the expected loss (�i). At the bandlevel, the model estimates the average number of defaults (mj) as the ratio of the total expectedloss in the band (�j) to the common exposure characteristic to the obligors from that band (�j).


By assumption, the distribution of the number of defaults in each band is of the Poisson type:

� �!k

emkjbandindefaultsofnumberPP

jmkj

j

�

�� k = 0, 1, ...

Further, the model employs statistical tools to estimate the distribution of loss at the portfoliolevel. Using the properties of probability-generating functions, the model estimates recursively theprobabilities that the portfolio loss reaches values expressed as multiples of the unit of exposure.

For our portfolio, the analytical risk assessments at the asset and band level are contained inTables 11 and 12.

Table 11: Risk Assessment at the Asset Level According to the CreditRisk+ Model

Asseti

Exposure(billion CZK)

Commonexposure (�i�)

Common exposurerounded to multiples of

CZK 1 billion (�i)

Probability ofdefault

(Pi)

Expected loss(�i=�i�� Pi)

1 28.92 28.92 29 0 0.002 28.92 28.92 29 0.0006 0.023 28.92 28.92 29 0.0018 0.054 28.92 28.92 29 0.0018 0.055 28.92 28.92 29 0.0106 0.316 28.92 28.92 29 0.0106 0.317 28.92 28.92 29 0.0106 0.318 28.92 28.92 29 0.052 1.509 28.92 28.92 29 0.052 1.5010 28.92 28.92 29 0.052 1.5011 28.92 28.92 29 0.052 1.5012 28.92 28.92 29 0.052 1.5013 28.92 28.92 29 0.052 1.5014 28.92 28.92 29 0.052 1.5015 28.92 28.92 29 0.052 1.5016 28.92 28.92 29 0.052 1.5017 28.92 28.92 29 0.052 1.5018 28.92 28.92 29 0.052 1.5019 28.92 28.92 29 0.052 1.5020 28.92 28.92 29 0.052 1.5021 28.92 28.92 29 0.052 1.5022 21.45 21.45 22 0.0106 0.2323 21.45 21.45 22 0.052 1.1224 20.38 20.38 22 0.052 1.1225 21.45 21.45 22 0.1979 4.2526 18.19 18.19 19 0.052 0.9527 18.19 18.19 19 0.1979 3.6028 18.19 18.19 19 0.1979 3.6029 13.49 13.49 14 0.1979 2.6730 13.49 13.49 14 0.1979 2.67



Four different bands have thus been obtained, with rounded common exposures of 14, 19, 22 and29. Table 12 illustrates the partition of the portfolio in bands and the risk characteristics of eachband.

Table 12: Band Partition and Risk Assessment at the Band Level According to the CreditRisk+Model

Band

j

Roundedcommonexposure

in Band j (�j)

Numberof

obligorsin Band j

Expectedloss in

Band j (�j)

Expectednumber of

defaults in Bandj (mj=�j/�j)

Probability-generatingfunction for Band j

1 14 2 5.340 0.381 exp(-0.381+0.381z14)

2 19 3 8.147 0.429 exp(-0.429+0.429 z19)

3 22 4 6.704 0.305 exp(-0.305+0.305z22)

4 29 21 22.092 0.762 exp(-0.762+0.762z29)


For each band a probability-generating function is given by

� � � � ��

�

��

��

�

�

�

��

0k

zmmkmk

j

0k

k

0n

njj

jjjj

j

j ez!k

emzdefaultskPznLPr)z(G

ννν .

The probability-generating function for the entire portfolio is the product of the individualprobability-generating functions displayed in the last column of Table 17. In this particularexample we get

29221914

jm

1jj

m

1jj

z762.0z0.305z4290.z0.3811.877zmm

ee G(z) ��

��

�

��

�

�

�� .

To derive the probabilities that loss equals multiples of the unit of exposure, CreditRisk+constructs a recurrence relationship:

j

j

n

n.t.sj

jjn P

nm

P��

��

��

�

that starts with the probability of no loss:

153.0e e e Loss) P(No P 1.877-m

m-0

m

1j´j

��

�

��

�

The resulting loss distribution in the case of our portfolio is shown in Figure 2.


0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0 10 20 30 40 50 60 70 80 90 100

110

120

130

140

150

160

170

180

Loss

Probability

Figure 2: Loss Distribution Based on the CreditRisk+ Model


In CreditRisk+, economic capital is given by the difference between the p-percentile and theexpected mean of the loss distribution.

Economic Capital = p-percentile – Expected Loss

Applying the CreditRisk+ approach to our portfolio, the estimation of risk capital at differentconfidence levels ended up with the values expressed below.

1%-percentile 5%-percentile Expected loss 99% Capital 95% Capital

133 101 42.18 90.82 58.82


7. Monte Carlo Simulation with Floating Lending Rates and Fixed ForwardZero Curves

We assume that the mechanism of future changes in lending interest rates is as follows:

te100r

1100r

1 tgt ��

��

�

�

��

�

��

�

Here σ is a normally distributed random variable, so the exponential follows a log-normaldistribution.In this formulation, interest rates on loans preserve the mark-ups that capture obligor-specific riskor liquidity premia (already incorporated in the old interest rates), but also contain a randomcomponent reflecting uncertainty related to the future course of the money market rate (PRIBOR).The random component does not vary across obligors. In our simulations it was obtained as arandom draw from a log-normal distribution. The parameters (mean and standard deviation)describing the log-normal distribution were estimated on actual Czech data (1Y PRIBOR over theyear 2000, a period of relative rate stability and no monetary policy changes).

7.1 Monte Carlo Simulation with Floating Interest Rates

A total number of 10,000 random draws from the standard log-normal distribution wereperformed to determine the random component of the PRIBOR and thus the loans’ interest rates.Due to the floating nature of the interest rates, the valuations of each asset and of the portfoliovaried in line with the particular values of the random draws. A total number of 10,000 portfoliovalues at the end of the risk horizon were thus obtained. Figure 3 shows the relative frequencies ofthese values at the year-end.

Figure 3: The Empirical Distribution of the Portfolio Value with Floating Interest Rates

Source: Own computation.Note: The horizontal axis shows non-overlapping intervals that cover the entire range of the estimated

portfolio values, and the vertical axis shows the frequencies with which the portfolio values fellinto those intervals. The descriptive statistics to the right are the same as in Fig. 1.

The estimation of economic capital in this case is given in Table 13.

0

200

400

600

800

1000

1200

600 700 800 900 1000 1100


Mean 848.3348Median 845.5493Maximum 1167.615Minimum 554.5349Std. Dev. 81.97142Skewness 0.195979Kurtosis 3.005247


Portfolio values

Frequency


7.2 Monte Carlo Simulation with Floating Interest Rates and Stochastic Forward ZeroCurves

The next four cases retain the assumption that floating interest rates were charged by the bank onits loans. Additional uncertainty is added with regard to changes in forward zero curves (thediscount factors entering the loan valuations) over the one-year period. We analyzed the impacton the bank’s need for economic capital under the following changes in the forward zero curves:upward translation, downward translation, clockwise rotation and counter-clockwise rotation.Rotations of the forward zero curves are around the point determined by the two-year forwardrate. These changes are illustrated in Figure 4.

Figure 4: Simulated Changes in the Forward Zero Curves

a) Upward translation b) Downward translation

c) Upward (counterclockwise) rotation d) Downward (clockwise) rotation

We assumed that these changes reflected subjective expectations concerning the evolution of theCzech forward zero curves over a one-year period. They became rating class-specific by addingthe US forward spread (the difference between the US rating class-specific and the US forwardzero curves). We incorporated these changes into the forward zero curves in the model accordingto the formula:

Forwardzerocurves

Year4Y3Y2Y1YYear

Forwardzerocurves

4Y3Y2Y1Y

Forwardzerocurves

Year4Y3Y2Y1Y

�

Forwardzerocurves

4Y1Y 2Y 3Y Year


4,3,2,1t,e100s

1100f

1100f

1 tgtt

gt ��

��

�

�

��

�

��

��

�

�

��

�

��

� (2)

Here ft is the original Czech forward zero rate at year t, sgt is the spread in forward zero rates

characteristic of the g-th rating class at time t, and �t is a random draw from the normaldistribution (thus e�t is log-normally distributed).

The proposed changes in the forward zero curves were captured in the model by consideringparticular random variables �t in (2):

,�� t t=1,2,3,4, for an upward translation,

,�� t t=1,2,3,4, for a downward translation

εµφεµφεµφ �� 2,, 431 , for an upward rotation

εµφεµφεµφ �� 2,, 431 , for a downward rotation.

The overall effect of an upward shift in the forward zero curve is a decrease in the present valueof all loans in the portfolio. If the bank heavily discounts the future, the opportunity cost ofgranting loans increases, since alternative assets may provide higher returns in the future.Accordingly, the present value of the cash flows accrued from the loans is lower compared withthe case where the forward zero curves remained unchanged. The effect of a downward shift ofthe forward zero curve is the opposite of the one mentioned above. Rotations of the forward zerocurve affect assets’ valuations depending on maturity. For example, the counter-clockwise(upward) rotation discounts assets with a short maturity less and assets with a long maturity more.Therefore, the valuation of the portfolio is very sensitive to the portfolio composition. If moreassets fall into the long maturity category the present value of the portfolio tends to fall, while ifthey fall into the short maturity category the present value of the portfolio tends to increase.

In all previous formulations of the �t distribution, the parameter � determined the magnitude ofthe deterministic change in forward zero curves and � added random deviations. We wantedchanges in the deterministic part of the discount factors not exceeding one percent (thus, if for agiven maturity the forward zero rate was 3.4%, we wanted it to deviate upward to 4.4% only).This assumption implied a value for � of 0.01. In each case, � was assumed to follow a normal

distribution with mean 2

2�

� and standard deviation �, so that the mean of the log-normally

distributed factor e� is equal to 1. Under these conditions � became normally distributed withmean �+m. The standard deviation � of � was estimated by computing the standard deviation ofthe log(1+1Y PRIBOR/100) variable using daily observations over the year 2000 after removingthe trend. The same estimated values of the parameters � and � were used in all cases of randomchanges in forward zero curves.

We performed Monte Carlo simulations containing 10,000 scenarios that simultaneouslyaccounted for random changes in interest rates and forward zero curves. Shown next are the


portfolio value distributions and the estimates of economic capital based on simulations thatincorporated the proposed changes into the forward zero curves.

Figure 5 displays the portfolio value distributions when the four FZC change cases discussedabove are compared.

Figure 5: Portfolio Distributions under Different Assumptions Regarding Changes in FZC


Economic capital estimations at different confidence levels are displayed in Table 13. Thedownward translation and the clockwise rotation of the forward zero curves impose the highestrequirements of economic capital in this particular example. However, economic capital seems toconverge towards the fixed forward zero curves (with floating lending rates) case when theconfidence level is reduced.

Table 13: Capital Requirements Assuming Different Changes in Interest Rates and ForwardZero Curves (fzc)

10,000 random draws 1%-percentile

5%-percentile

Mean 99% Ec.capital

95% Ec.capital

Fixed interest and fixed fzc 767.90 796.62 845.78 77.89 49.16Floating interest and fixed fzc 669.65 718.48 848.33 178.69 129.85Upward translation of fzc 655.15 705.70 834.25 179.11 128.56Downward translation of fzc 672.44 724.32 856.94 184.50 132.62Counter-clockwise (upward) rotation offzc

668.83 722.92 839.14 170.31 116.22

Clockwise (downward) rotation of fzc 668.83 722.92 853.01 184.18 130.09Source: Own computation.

0

200

400

600

800

1000

1200

[500,

520)

[540,

560)

[580,

600)

[620,

640)

[660,

680)

[700,

720)

[740,

760)

[780,

800)

[820,

840)

[860,

880)

[900,

920)

[940,

960)

[980,

1000

)

[1020

, 104

0)

[1060

, 108

0)

[1100

, 112

0)

[1140

, 116

0)

[1200

, 122

0)

Up Translation Dow n Translation Up Rotation Dow n Rotation


8. A Structural Model of Risky Debt with a Random Default Arrival

We next give an outline of a model of risky debt, its valuation and the resulting economic capitalrequirements, going along the “structural” lines of the original KMV model and its ramifications.The term “structural” means that we make the default explicitly dependent on loan and obligorcharacteristics. However, we borrow an additional element from the so-called “reduced-form”models of default (for a survey of both types of model, see Bohn, 1999), by working with adefault process in Poissonian form. This technique was introduced by Jarrow and Turnbull, 1995.We follow the variant utilized by Madan and Unal, 1998, 1999, in that the default event arrivalrate becomes a function of the same obligor fundamentals as the ones that drive the asset prices.However, this principle is developed in a way to establish a link between the default process ofthe abstract reduced-form models and the empirics inspired by the Expected Default Frequencynotion of KMV. The proposed model allows one to deal with a loan portfolio with correlateddefaults in a natural way.

8.1 Definitions

Consider a portfolio of n loans issued by n different companies (obligors). Loan j pays a couponj

tc at time t and the coupon plus the face value Fj at maturity Tj (t=1,…,Tj). The value of the loan

to firm j at time t is denoted by jtV . Then, Bt=�

�

n

j

jtV

1

is the value of the loan portfolio. We are

interested in finding B0.

There are N assets traded in the market, with prices ktP at time t (k=1,…,N). These assets

represent all sources of aggregate uncertainty in the economy independent of the actions ofobligors defined above. In this sense, the financial markets outside the considered borrower set arecomplete. These uncertainty factors will be represented by random variables Sk. By S, we denotethe vector with components Sk. Let S be a Markov process with a given transitional probability.

The loans are risky. If firm j generates period t-cash flow )( tjj

t SfY � net of all other debtservice obligations, then the probability of default j

t� on the loan is an inverse function of thedifference j

tj

t CY � : )()( jjj CYS �� . Here, jt

jt cC � if t<Tj, and jj

tj

t FcC �� if t=Tj.Hence, the variable driving the default rate in our model is an analogue of the distance-to-defaultmeasure used in KMV. One should think of � as approaching unity when the distance to defaultfalls to minus infinity, and approaching zero when it increases to plus infinity.

The space of the random events in our model is formed by pairs �=(s,b), where s is a realizationof S and b=[b1,…,bn], bj=S if there is no default (survival) and bj=D if the firm defaults ontheloan. The arrival fact of the default event itself is assumed to be independent of S, i.e. only theprobability value of the default is S-dependent, through the cash flow variable Yj.

For each loan, there is collateral that is tradable and depends on the same sources of uncertainty asthe basic assets. That is, the collateral price for loan j is equal to Zj=�j(�). If the loan defaults, thebank seizes the collateral, i.e. receives the value of Zj.

There are two important cases to be distinguished with regard to the collateral prices. Onepossibility is to allow �j to depend on both s and b. That is, this collateral is worth different


amounts depending on whether the debt has defaulted or not. This would be the case if there werea separate structural factor behind the realization of b, correlated with the market risk factors S.The same factor should be responsible for the value of the collateral. This situation would allowloan j to be priced in accordance with the risk-neutral valuation principles (see an example of sucha valuation in Derviz and Kadlčáková, 2002, Section 3). It may occur when the collateral is veryobligor-specific.

However, an equally legitimate case is that of the collateral being totally unrelated to theoperation of the firm (e.g. securities in its investment portfolio). Then Zj=�j(s) and a unique risk-neutral valuation of the loan is impossible. In that case, one must resort to pricing techniquesbased on explicit individual portfolio optimization. This is done next.

8.2 The Individual Loan and the Portfolio Value

Let us consider an optimizing investor in discrete time who decides upon allocating his/her wealthbetween the existing marketable assets, i.e. the n traded securities, the n collateral assets and the ncompany loans (all defined above). Let this be an optimization problem under uncertainty indiscrete time with transaction costs, of the type discussed in Derviz, 2000.

Let gj be the stream of coupons/dividends paid out by the basic security j, and hj the same thingfor the collateral security j. These values are unknown at the beginning of each period, when theinvestor makes the portfolio allocation decisions. Define yj as the current yield on the basicsecurity and zj as the current yield on the collateral:

jt

jt

jtj

t PPgy 11

11 ��

�

�

�� , jt

jt

jtj

t ZZhz 11

11 ��

�

�

�� .

it+1 will denote the risk-free short rate between periods t and t+1. The period utility function of theinvestor is a function of two variables: current cash holdings and the dividend rate withdrawnafter the investment strategy gains are realized: u=u(x,��The �-dependence is of the standardInada form, and the x-dependence is strictly increasing and concave, with the limit -∞ when xgoes to -∞, for every �. If the time preference rate of the investor is ��(0,1), the pricing kernel(stochastic discount factor, see Campbell et al., 1997, or Cochrane, 2001) is given by

),(),( 111

tt

tttt xu

xu�

��

�

� ��

�� , ��

�

��

�

�

tk

kkt

1 , �>t, 1��tt

(subscripts by u denote partial derivatives).

The information available to the investor at time t consists of the trajectories of g, h, P and Z aswell as the default event realizations b, all up to time t. The no-default up to time t subset of theevent space for security j will be denoted by j

tN . Let the S-dependent statistics of survival, jtK�, ,

between times t and �t+1, be defined as


� ��

��

�

��

1, 1

tk

jk

jtK .

Now we apply the standard asset pricing theory results. The optimal investor behavior implies thefollowing asset pricing formulae (special cases of the discrete time consumption-based CAPM):

� � ,1)1( 11

��

� jt

ttt yE � � ,1)1( 1

1��

�

� jt

ttt zE � �

),(),(1)1( 1

1

tt

ttxt

ttt xu

xuiE�

�

�

��

� , (3)

��

��

��

�

�t

jtt

jt hEZ

�

�

� , (4)

� ��

��

��

�

jT

t

jjjtt

jt

jt hCNEZV

�

��

� . (5)

In view of our assumptions about the default arrival independence of S, equation (5) can berewritten in the form

� � � �jT

Tt

jTtt

T

t

jjt

jtt

jt

jt j

j

j

j

ZKEhcKEZV 11

1,*

1,*

�

�

�

�

��

�

��

��

�

��

�

�, (6)

where now the conditional expectation *tE is taken only with respect to the market-wide risk

factors. Thus, we have eliminated the firm-specific default events from the debt pricing formulae(5). Also note that the asset pricing equations (3) could be written with expectation *

tE instead ofEt from the outset.

Now one can utilize the previously made market completeness assumption to note that the valueof the individual loans and the loan portfolio as a whole can be calculated as soon as onereconstructs a formula for the pricing kernel �

t� from the pricing equations (3). Standardorthogonal basic decomposition methods known from numerical mathematics can be used to dothis. Examples in the literature include Ait-Sahalia and Lo, 2000, Jackwerth, 2000, and Rosenbergand Engle, 2002. The necessary inputs are the price and yield processes for the basic assets andcollateral assets, both available as market data.

8.3 Application to Economic Capital Calculations and Simulations

The model outlined above is likely to be free of certain deficiencies typical of the two standardones, whose application to economic capital calculations was described in detail in Sections 5–7.For instance, a counterintuitive negative dependence of the capital requirement on the marketinterest rate is an unfortunate feature of the way CreditMetrics works with the relation betweenthe debt value and the economic capital. That model does not have any link between interest ratesand the firm’s ability to repay the loan. On the contrary, our model is able to do so, because thenet cash flow Yj will usually be negatively related to the market rates of interest. Therefore, a


reduction of the latter (such as a downward translation of the forward zero curve) increases Yj andthereby reduces the default event rate.

Another important advantage of the model is its ability to handle correlated defaults in a naturalway. This is because the default correlation in the model is not an exogenously given property oran ad hoc assumption, but instead follows by construction from the dependence of the defaultrates on the common risk factors S.

A certain difficulty lies in the necessity to calculate the pricing kernel recursively for multi-periodloan contracts. However, the calculations themselves are routine and are based on well-developednumerical techniques, allowing one to apply relatively standard software.

Once the distribution of the loan portfolio market price has been calculated, it can be used toderive the bank-internal measure of economic capital. The latter will subsequently be comparedwith the prudential capital values derived directly from the statistics of S and b according toregulatory principles. The difference will tell us the degree of discrepancy between internal riskmeasurement and regulatory mechanisms of risk-based capital allocation by the bank.

Details of the numerical procedures used to simulate our artificial portfolio value according to thepresented model are given in Appendix A3.

Figure 6: Portfolio Value Arbitrage-Based (Risk-Neutral) Distribution According to thePricing-Kernel Loan Valuation Model

The corresponding economic capital requirements are given in Table 14.

Table 14: Capital Requirements According to the Pricing-Kernel Model

Riskless value of the portfolio Mean Economic capital832.86 747.89 84.97

0

2000

4000

6000

8000

0 2500 5000 7500 10000 12500 15000 17500


Mean 747.8904Median 227.5549Maximum 18628.39Minimum 0.052275Std. Dev. 1483.338Skewness 4.554015Kurtosis 31.46530


Frequency

Portfolio value


If one compares the above valuation with the one obtained by means of the CreditMetrics model(Section 5), the increase in the portfolio valuation coming from the pricing-kernel approach issubstantial. It is illustrated in Figure 7 below.

Figure 7: Portfolio Value Distributions Based on the Monte Carlo Simulation of CreditMetricsand the Pricing-Kernel Model

The reasons for this difference are twofold. First, the CreditMetrics valuation uses a correlationstructure taken over automatically from the representative equity prices of firms in the consideredobligor classes. This correlation structure is necessarily inaccurate, particularly in reflecting therole of systemic risk factors (cf. the comment on the market interest rate impact in Section 6). Ourmodel incorporates systemic factors directly, which means that macro-developments aretransmitted directly into the whole portfolio and change its value in accordance with intuition.Second, we have used a somewhat different collateral valuation in the pricing-kernel model thanin the CreditMetrics simulations. A collateralized loan in our understanding always pays out arecovery rate in case of default, even though this rate is subject to random fluctuations around theface value of the debt. On the contrary, our CreditMetrics exercise worked with an understandingthat was close to the Czech regulatory one, i.e. certain collateral categories were given zerorecovery rates. This happens even though the initial value of every item of collateral is equal tothe face value of the loan. Altogether, the market collateralized debt valuation in terms of standardasset pricing theory is naturally biased toward reducing the differences between default and no-default realizations of uncertain factors (the risk-neutral probability of an asset valueupside/downside shift is lower/higher than the actual probability). Therefore, this theory subsumessuch asset prices that agents become nearly indifferent between holding loans and collateral. Incomplete asset markets, the indifference is perfect. Here, we are working with non-traded loans,which generate market incompleteness, but the pricing-kernel valuation would still alleviate theloan-collateral demand discrepancy to the extent that the default events and collateral values aredependent on common systemic risk factors.

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

[0, 20

0)

[620,

640)

[700,

720)

[780,

800)

[860,

880)

[940,

960)

[1040

, 106

0)

[1120

, 114

0)

[1200

, 140

0)

[2000

, 250

0)

[4000

, 450

0)

[6000

, 650

0)

[8000

, 850

0)

[1000

0, 10

500)

[1250

0, 13

000)

[1700

0, 17

500)

CreditMetricsPricing Kernel Model


The basic properties of the pricing kernel-based valuation can be summarized as follows.A. Every loan value is dependent on the market attitude towards risk.B. The arbitrage-free (i.e. financial market-based) loan value distribution gives a higher weight

(prices higher) to recovery value payments of the collateral, and a lower one to the originalrisk adjusted-loan service payments, than the actual probability.

C. For each macroeconomic or other systemic risk factor, loans with default probability/cashflow negatively/positively correlated with this factor obtain a valuation increase from thatfactor. So, for instance, bank claims on exporters are valued higher when the exchange rate isincreasing (the national currency is depreciating).

These properties cannot be obtained from the CreditMetrics- or CreditRisk+-based valuation, afact which introduces considerable bias into their results. The KMV model probably works withthe same phenomena as described in this section. However, according to existing public sourceson that model, it does not overcome the structural vs. reduced-form modeling dilemma for privateborrowers. This is one of the main original contributions of the present pricing kernel-basedmodel.

9. Conclusion

The following table sums up the credit risk-related estimations of regulatory and economic capitalin all the cases considered. The impact of lending rate variations (market risk) is partially reflectedin some of the capital requirement estimations.

Table 15: Summary of Capital Estimations

Capital requirement1%-

percentile5%-

percentileMean 99%

Economiccapital

95%Economic

capital

Non VaReconomic

capital

Regulatory

capitalCreditMetrics 767.90 796.62 845.78 77.89 49.16CreditRisk+ (Loss) 133 101 42.18 90.82 58.82

Pricing-kernel model 747.8904 84.97Floating interest and fixedfzc

669.65 718.48 848.33 178.69 129.85

Upward translation of fzc 655.15 705.70 834.25 179.11 128.56Downward translation offzc

672.44 724.32 856.94 184.50 132.62

Counter-clockwise(upward) rotation of fzc

668.83 722.92 839.14 170.31 116.22

Clockwise (downward)rotation of fzc

668.83 722.92 853.01 184.18 130.09

NBCA – Standardizedapproach (January 2001)

51.84

NBCA – IRB approach(January 2001)

165.46

NBCA – Standardizedapproach (October 2002)

46.90

NBCA – IRB approach(October 2002)

44.79

Accordingly, the main conclusions of this study can be summarized as follows.

• The capital requirements according to the January 2001 NBCA consultative document andOctober 2002 QIS 3 technical guidance differed significantly. The newer guidance was lessdemanding on the capital level and made the use of the IRB approach more convenient to thebanks in the sense of the capital level required.

• The regulatory capital requirement resulting from the January 2001 NBCA guidance wasmuch lower in the NBCA’s standardized approach than in the IRB approach. Estimatedcapital was similar in the standardized approach and the modeling part when a 95%confidence level was selected in the models. Although at a 99% confidence level the modelsimposed higher capital requirements, they were still lower then the estimated capital in theIRB case of the regulatory requirements.

• Following the QIS 3 technical guidance from October 2002, the regulatory requirements ofthe two NBCA approaches were quite similar. The capital requirements according to bothregulatory approaches were lower than the level of capital required by the credit risk models,but the difference was not significantly higher.

• When floating interest rates and changes in yield curves were modeled, the estimate ofeconomic capital was generally higher. This result is intuitive, as increased uncertainty shouldgenerally impose higher levels of capital that banks must hold.

• The particular changes in the forward zero curves analyzed in this paper did not imposesignificantly different levels of economic capital than the case with stable forward zerocurves. The case when forward zero rates went downward (both translation and clockwiserotation) required slightly more capital, and this situation persisted at all levels of confidenceselected.

• The risky debt valuation obtained by conventional incomplete market asset pricing methodstended to generate higher loan values and reduce the economic capital requirements comparedto the other regulatory and model-based risk measurement methods. Therefore, the regulatormay encounter an effort on the part of the banks to treat different parts of the loans on theirbalances differently in terms of economic capital. The difference will go in the direction ofreducing capital allocation (and specializing collateral requirements) in those segments of theloan portfolio which exhibit strong correlation with traded risks.

• Asset pricing methods of risk measurement may lead to a better recognition of the role of thebusiness cycle and other systemic macroeconomic factors in economic capital determination.The pro-cyclical nature of many existing risk management procedures may therefore give wayto natural (and desirable) counter-cyclical adjustments.


References

AIT-SAHALIA, Y., AND A. LO (2000): “Nonparametric Risk Management and Implied RiskAversion”, Journal of Econometrics 94, 9–51.

BARNHILL, T., AND W. MAXWELL (2002): “Modeling Correlated Market and Credit Risk inFixed Income Portfolios”, Journal of Banking and Finance 26, No. 3, 347-374.

BASEL COMMITTEE ON BANKING SUPERVISION (January 2001): The New Basel Capital Accord,Consultative Document, Bank for International Settlements.

BASEL COMMITTEE ON BANKING SUPERVISION (January 2001): The New Basel Capital Accord:An explanatory note, Bank for International Settlements.

BASEL COMMITTEE ON BANKING SUPERVISION (25 June 2001): Update on the New Basel CapitalAccord, press release, Bank for International Settlements.

BASEL COMMITTEE ON BANKING SUPERVISION (13 December 2001): Progress towardscompletion of the new Basel Capital Accord, press release, Bank for InternationalSettlements.

BASEL COMMITTEE ON BANKING SUPERVISION (10 July 2002): Basel Committee reachesagreement on New Capital Accord issues, press release, Bank for International Settlements.

BASEL COMMITTEE ON BANKING SUPERVISION (October 2002): Quantitative Impact Study 3,Technical Guidance, Bank for International Settlements.

BASEL COMMITTEE ON BANKING SUPERVISION (1 October 2002): Quantitative Impact Survey ofNew Basel Capital Accord, press release, Bank for International Settlements.

BOHN, J. (1999): “A Survey of Contingent-Claims Approaches to Risky Debt Valuation”,Working Paper, Haas School of Business, U.C. Berkeley (June).

BUCHTÍKOVÁ, A. (2002) Přehled průměrných finančních podílových ukazatelů firem (1999–2000), CNB.

CAMPBELL, J., A. LO, AND C. MACKINLAY (1997): The Econometrics of Financial Markets,Princeton, NJ: Princeton Univ. Press.

COCHRANE, J.H. (2001): Asset Pricing, Princeton, NJ: Princeton Univ. Press.CREDIT SUISSE FIRST BOSTON (1997): Credit Risk+ – A Credit Risk Management Framework.CROSBIE, P. (1999): Modeling Default Risk, KMV Corporation.DERVIZ, A. (2000): “Monetary Transmission and Asset-Liability Management by Financial

Institutions in Transitional Economies – Implications for the Czech Monetary Policy”,Focus on Transition I, 30–66 (Oesterreichische Nationalbank, March).

DERVIZ, A., AND N. KADLČÁKOVÁ (2002): “Strategic Outcomes of Credit Negotiation andPrudential Capital Regulations”, CERGE-EI Discussion Paper No. 82 (March).

DUFFIE, D. AND K. SINGLETON (1998): Modeling Term Structures of Defaultable Bonds.Stanford University.

GUPTON, G., C. FINGER, AND M. BHATIA (1997): Credit MetricsTM – Technical Document.JACKWERTH, J. (2000): “Recovering Risk Aversion from Option Prices and Realized Returns”,

Review of Financial Studies 13, 433–451.HOU, Y. (2002): Integrating Market Risk and Credit Risk: A Dynamic Asset Allocation

Perspective, Mimeo, Yale Univ., Dept. of Economics (November).


JARROW, R.A., D. LANDO AND S.M. TURNBULL (1997): “A Markov Model for the TermStructure of Credit Risk Spreads”, Review of Financial Studies, 10(2), 481–523.

JARROW, R.A. AND S.M. TURNBULL (1995): “Pricing Derivatives on Financial Securities Subjectto Credit Risk”, Journal of Finance, 50(1), 53–85.

MADAN, D., AND H. UNAL (1998): “Pricing the Risks of Default”, Review of DerivativesResearch 2, 121–160.

MADAN, D., AND H. UNAL (1999): “A Two-Factor Hazard-Rate Model for Pricing Risky Debtand the Term Structure of Credit Spreads”, Working Paper, Smith School of Business,Univ. of Maryland (November).

ROSENBERG, J., AND R. ENGLE (2002): “Empirical Pricing Kernels”, Journal of FinancialEconomics 64, 341–372.

SELLERS, M., AND A. DAVIDSON (1998): Modeling Default Risk: Private Firm Model, KMVCorporation.


Appendix

A.1 Generating the Unobservable Characteristics of the Artificial Bank Loan Portfolio

A1.1 Predictors of Default

We applied Moody’s guidelines (see Section A.2 for a methodology outline) to obtain ratings forthe firms present in our pool of Czech bank customers. Based on accounting data for 1997, weconstructed financial ratios and tested their power in predicting default in 1998. In this sense, weperformed a series of univariate probit regressions containing as dependent a variable taking thevalue 0 if a firm defaulted and 1 if a firm did not default between 1997 and 1998. We looked forcandidates for the explanatory variables belonging to five out of the six categories proposed bythe RiskCalc model: profitability, leverage, liquidity, size and activity10. The variables that weretested within each category are displayed in Table A1. Marked with a star are those variables thatwere found significant at least at the 90% confidence level in the univariate probit tests.

Table A1: Explanatory Variables of the Univariate Probit Regressions

Category VariableProfitability - EBIT/Assets

- Net Income/Equity- Net Income/Assets*- Operating Profit Margin

Leverage- Total Debt/Total Assets*- Debt Service Coverage Ratio- Bank Debt Ratio*- Equity Ratio

Size - Total Assets/PPI97- Sales/PPI97

Liquidity - Quick Ratio*- Cash/Total Assets- Short Term Debt/Total Debt- Current Ratio*- Cash Position Ratio*- Net Indebtedness*

Activity - Sales/Total Assets*- Inventories/COGS- Interest Expense/Sales

The only significant variable (at the 90%, but not at the 95%, confidence level) in the profitabilitycategory was Net Income/Assets. In the activity category we also found only one significantvariable (Sales/Total Assets). In the leverage category two variables were significant: Bank DebtRatio and Total Debt/Total Assets. The Bank Debt Ratio dominated the Total Debt/Total Assetsvariable, so the former was the only variable chosen in the leverage category. The majority of thevariables in the liquidity category were significant. To select only one or two we applied the 10 Measures of growth such as sales or asset growth were not used, because only slightly more than 300 firmswere recorded in 1996 out of the 606 firms considered between 1997 and 1998. Thus, the use of lagged variableswould have reduced the number of observations, restricting the available information about defaults and thedegrees of freedom of the regression.


technique proposed by Moody’s. Starting with the most significant one (Cash Position Ratio) weadded liquidity-representative variables in this probit regression until they proved insignificant orhad a counterintuitive sign. Actually, none of the remaining variables proved more powerful thanthe Cash Position Ratio, so this was the only variable chosen. Size variables showed nosignificance in relation to default, so we did not include any variable from this category.

Overall, the variables that were found significant in the simple probit tests were NetIncome/Assets, Bank Debt Ratio, Sales/Total Assets and Cash Position Ratio. Consequently, weran a probit regression using all these four explanatory variables. The outcome is displayed inTable A2.

Table A2: The Probit Regression Outcome Based on four Explanatory Variables

Variables ( 1997jX ) Coefficient ( 1997

jc ) t-statistics

Constant 1.19 7.43Net Income/Assets 0.006 1.06Cash Position Ratio 0.83 2.62Bank Debt Ratio -0.46 -1.45Sales/Total Assets 0.114 1.85

A1.2 Probability of Default Estimation

We used the 1997–1998 database as the reference data set. As our bank data set extended until2000 and because we wanted to use the most recent information available, we assumed that thestructural relationship between default and the variables given in Table 2 was stable over time andtherefore applicable in 200011. Based on this assumption it was possible to estimate the one-yearprobability of default for each firm in the data set for 2000 according to the relation:

)(4

1

2000199719970

2000��

��

jijj

Defaulti XccP .

Here Φ is the standard normal cumulative distribution function, cs are the estimated coefficientsbased on the 1997 data (see Table 2) and Xij are the four financial ratios computed for each of the1,035 firms present in the 2000 data set. These predictions of the annual default rates are furtherused to assign dot.PD-like ratings.

A1.3 Types of Collateral and Recovery Rates

The type of collateral was assigned randomly to the individual assets by drawing a randomnumber from the distribution displayed in Table 7. The support of this distribution consists ofnumbers from 1 to 6. They codify the types of collateral accepted by Czech banks (1=bank 11 Although this assumption is not completely out of the question, we had to rely on it due to the unavailability ofmore reliable default information in 2000.


guarantee, 2=cash, 3=securities, 4=commercial real estate, 5=other, 6=no collateral). Theprobabilities with which the choices were made mirrored the ratios of the volume of the individualtypes of collateral offered to the total volume of classified assets. To calculate these ratios, weused the information about the structure of classified assets of the Czech banks’ portfolios at theend of 2000 as given by CNB sources. We assumed that each asset in our simulated portfolio waseither fully covered by specific collateral or not covered at all.

Due to unavailability of own data, the recovery rates of the individual assets were computed,based on their collateral quality, as

Recovery rate = 100/(1 + He + Hc),

where He and Hc are the “haircut” coefficients corresponding, respectively, to the loan and thecollateral volatility. They reflect subjective assessments that took into account the collateralquality (see Table A3 below). The computation was based on the NBCA methodology for settingthe value of adjusted collateral (from January 2001, cf. Subsection 5.1).

Table A3: Collateral Type Distribution and Haircut Coefficients

Collateral Type Share Coefficient1. bank guarantee 0.368 0.032. cash 0.0125 0.03. securities 0.006 0.064. commercial real estate 0.183 0.25. other 0.12 0.56. no collateral 0.31 X


A.2 RiskCalcTM for Private Companies: Moody’s Default Model

Moody’s RiskCalcTM model predicts private firms’ default probabilities based on selectedfinancial indicators. Unlike public firms, which are quoted and regularly traded on a stockexchange, private firms’ securities are infrequently traded. This fact restricts the access to marketinformation for these firms, implying that such information can be hardly incorporated into adefault model. There are other differences between public and private firms that necessitate aspecial default model for private firms. As documented in Moody’s proprietary databases, privatefirms typically have smaller size and higher retained earnings, and maintain more current debt andbelong to the medium range in terms of asset size. These and other factors would cause ananalysis of the default behavior based on financial statement data to produce disparate resultswhen applied to public and private enterprises. Comparative studies performed by Moody’s onprivate and public companies confirmed this outcome.

In devising, implementing and validating its default model for private firms, Moody’s used threelarge databases:

� A proprietary database called Moody’s Credit Research Database (CRD), containingaccounting and default-related information on private, middle-size market firms.


� Moody’s Default Database (MDD), containing information on US and Canadian public firmsthat have defaulted since 1980.

� This latter database was extended with financial data for non-defaulted public firms availablein Compustat’s Research Insight database.

At the time the technical document was published (May 2000) the content of these databasesallowed one to obtain a reliable construct and evaluation of Moody’s RiskCalc model. Someaspects relating to these data sets are displayed in Table 1.

Table A4: Characteristics of the Data Sets Used by Moody’s

Database SummaryTime Span Number of Firms Number of

DefaultsNumber ofFinancial

StatementsPrivate Firms 1989–1999 24,718 1,621 115,351Public Firms 1980–1999 15,805 1,529 130,019

Source: Moody’s RiskCalc™ for Private Companies.

Moody’s model for rating private firms has been also applied in a number of European countries:Belgium (2002), France (2002), Germany (2001), the Netherlands (2002), Portugal (2002), Spain(2002) and the United Kingdom (2002). Table 2 presents the main features of the data sets used inthese countries. The last row contains similar information about the data set we used in the CzechRepublic. Our data is poorer than the data sets used in other European countries. However, ourgoal was not to maintain the complexity and robustness of Moody’s rating process, but to use itsmethodology to build our own test portfolio. For this objective, the data set we used was relevant.Recently, the Czech National Bank officially launched a Central Register of Loans, in cooperationwith the Czech Banking Association. The register will enable banks to exchange information onloans provided to businesses. Commercial banks had been calling for the creation of a creditregister for several years, but its instigation was not possible until the private data protection issuehad been resolved.

Table A5: Characteristics of the Data Sets Used by Moody’s in Europe and of the Data SetUsed in the Case of the Czech Republic

Country Time Span Number ofFirms

Number ofDefaults

Number of FinancialStatements

Belgium 1992–1998 102,594 6,658 523,057France 1990–1999 253,268 25,229 1,323,754Germany 1987–1999 24,866 1,485 111,427Netherlands 1992–1999 19,327 436 79,696Portugal 1993–2000 18,137 416 69,765Spain 1992–1999 140,790 2,265 569,181UK 1989–2000 64,531 4,723 283,511Czech Republic 1997–1998 606 51 606

Source: Moody’s RiskCalc™ for Private Companies.


Any default prediction analysis must start with a definition of default. Moody’s formal treatmentof a default event is triggered by any of the following changes in the loan status of a bank client:

� a loan is 90 days past due;� the credit is written down (it is placed in the regulatory classifications of substandard,

doubtful or loss);� the loan is classified as non-accrual; or� bankruptcy is declared.

Moody’s rating methodology comprises the following basic steps:

� Single Factor Analysis: the aim is to select the most significant input variables for thesubsequent analysis and to describe the shape of the relationship to default of each relevantfactor.

� Model Specification and Estimation: those variables which were found to be powerfulpredictors of default in single factor tests are combined into multinomial logit or probitmodels. Typically, such models deliver scores and model-based probabilities of default foreach bank client.

� Mapping: the estimated scores and default probabilities can be mapped into standard,alphanumeric Moody’s ratings.

A2.1 Single Factor Analysis

Without a structural model dictating the choice of the explanatory variables to be included in themodel, variable selection follows a forward selection process. The selection of the explanatoryvariables starts by constructing financial ratios representative of the following risk categories:profitability, leverage, firm size, liquidity, activity and growth. In each category the model selectsthe most powerful predictors of default based on univariate tests. Subsequently, it adds lesssignificant variables to the model until they prove no additional predictive power in multivariatetests. As for the number of variables to be included in the model, Moody’s has reached theconclusion that seven represents the optimal choice. Without inflicting this conclusion on anymodel application, Moody’s warns against the tendency to overfit the model by including toomany variables. While the inclusion of additional explanatory variables always increases the fit ofthe regression, the risk of multicolinearity is also present. Multicolinearity has negativeconsequences for the reliability of the estimated coefficients and deteriorates the out-of-sampleperformance of the model.

Moody’s search for powerful predictors of default starts by defining the broad categories in whichto look for candidate financial ratios. The impact that each of these categories is likely to have ondefault and the variables chosen by Moody’s in each category are listed below.

� Profitability: normally, a firm that makes profits (that is, covers its own costs by ownrevenues) is able to honor its liability contracts and does not default. Thus, the higher theprofitability of the firm the lower the default risk is expected to be. The RiskCalc modelexamined the following variables in this category: EBIT/Assets, Net Income/Equity, (NetIncome - Extraordinary Items)/Assets and Operating Profit Margin12.

12 EBIT = Earnings Before Interest and Taxes. Operating Profit Margin = (Sales - Cost of Goods Sold, General andAdministrative Expenses) / Sales.


� Leverage: expresses the ability of the firm to pay debt from its own sources. More indebtedfirms pose a higher risk to lenders, so leverage is positively related to default. The variablesconsidered by Moody’s in this category were: Total Liabilities/Tangible Assets, TotalDebt/Total Assets, Total Liabilities/Total Assets, Total Debt/Net Worth and Debt ServiceCoverage Ratio (EBIT/Interest).

� Size: bigger firms are usually better diversified (that is, are less subject to idiosyncratic risk)and hold a stronger market position (a feature used in underwriting). Normally, size isnegatively related to default. The size variables considered were real sales or assets (nominalvalues divided by the CPI) and a measure of the market value of the firm’s equity.

� Liquidity: measures a firm’s ability to cover current (short-term) debt using its own currentassets. While low liquidity may act as a signal of default (since default is the result of a lack ofliquidity) it can also reflect active investment by the firm (cash and equivalents are thusquickly invested in other assets). For this reason, Moody’s questioned whether liquiditypredicts default in a timely manner, since signaling default one month in advance has lessrelevance for creditors than signaling it one or two years in advance. The variables included inthis category were: Quick Ratio, Working Capital/Total Assets, Cash/Total Assets, ShortTerm Debt/Total Debt and Current Ratio.

� Activity: it is assumed that a firm with a higher turnover has a higher likelihood of paying itsfinancial obligations. Thus, the variables in this category are expected to be negatively relatedto default. Among the variables included here one can mention the following: AccountsReceivable/COGS, Accounts Receivable/Sales, Sales/Total Assets, Inventory/COGS andAccounts Payable/COGS (COGS=cost of gold sold).

� Growth: the variables considered in this group are sales and asset growth over a one-year ortwo-year period. Growth is the only risk factor that displays a strong non-monotonerelationship with default. While firms with low sales growth have bad financial prospects,firms that grow too quickly are also risky because such growth is often fueled by increasedexternal debt. Even though the model builders recommend giving preference to monotonerelationships, growth was found to be a significant predictor of default and was thereforeincluded.

Variable selection proceeds on the basis of empirical tests assessing the statistical power of theindividual variables. By statistical power, Moody’s means the ability of individual financial ratiosto distinguish defaulted firms from non-defaulted. Statistical power is examined on the basis ofdefault frequency Fig.s and power curves.

Another aspect addressed within the single factor analysis is a search for those functional formswhich best describe the nonlinear relationships between default and the individual financialindicators (which are in general nonlinear, especially in the case of powerful predictors of default.The transformations are not known a priori (exponential, polynomial, etc.). In handling thisproblem, Moody’s turned to a nonparametric approach. Nonparametric regression extends thetraditional regression technique by avoiding any special assumptions about the functional form ofthe regression curve and about the statistical distributions of the error terms. The shape of theregression curve is estimated simultaneously with the unknown coefficients. Nonparametrictechniques are used to shape the individual relationships between financial ratios and futuredefaults. The result is a set of transformations that closely resemble the individual defaultfrequency representations when displayed in a Fig.ical form. The transformed variables are thennormalized and used as inputs in the multivariate probit or logit model.


A2.2 Model Specification and Estimation

RiskCalc combines the normalized transformed variables into the estimation of a multivariateprobit or logit model. The dependent variable in such a model is a dichotomous variable Y, takingthe values 0 and 1 in the case of default and non-default respectively.

The causality relationship between the dependent variable Y and the n explanatory variables Xj

(normalized transformed financial ratios) is captured indirectly through a latent variablespecification. Suppose that Y* is a latent variable whose rapport with Y is given by

��

��

�

��

�

�

0Yif,1

0Yif,0Y

i

ii .

Y* is assumed linearly related to Xj as in

ij,i

n

1jj0i XccY ��

�

� .

Here ε is the error term, assumed to follow a standard normal distribution in the probit model anda logistic distribution in the logit model.

The probability that firm i defaults given a certain realization of predictors Xi,j can be expressed as

� � � � � � ,XccPX|0YP0YPdefaultsiFirmP j,i

n

1jj0ij,iii �

�

�

�

��

�

�

��

�

�

��

�

��

�

�

or

� ��

�

�

��

�

�

��

�

�

��

�

��

�

j,i

n

1jj0 XccFdefaultsiFirmP (A1)

Here F is the cumulative distribution of the error terms, i.e. the cumulative standard normal

distribution in the probit model (Ф in the usual notation) and � � x

x

e1exF�

� in the logit model.

The coefficients c0, ci, i=1,n are estimated using a maximum likelihood method. This method iscommonly available in any statistical and econometric software package. Once the coefficients ofthe regression are estimated, the default probabilities for each firm are computed according to(A1).

The output of the multinomial probit or logit model is a set of raw estimates of the defaultprobabilities for the firms in the sample. Based on look-up tables (which also have a proprietarycharacter) these initial estimates are adjusted to produce Expected Default Frequency (EDF)-likedefault probabilities. These estimates are the final output of the RiskCalc used further to mapfirms into alphanumeric ratings.

A2.3 Mapping to Alphanumeric Ratings

Standard Moody’s ratings, known under their alphanumeric representation as Aaa, Aa1,…, C,capture the default risk of long-term corporate bonds. In its bond default study13 Moody’s 13 Available on the joint Moody’s and KMV web site at http://www.moodyskmv.com.


provides the theoretical background for computing historical average bond default rates for eachrating category. By comparing a firm’s predicted default probability with these rating class-specific bond default rates, Moody’s constructs dot-PD ratings for private firms (Aaa.pd,…,C.pd).While not entirely equivalent, bond ratings and dot-PD ratings are related, since each represents aform of corporate debt. Mapping corporate default rates into bond ratings can prove useful inseveral circumstances, including credit risk assessments performed on bank loan portfolios. Inputdata in credit risk models, such as migration matrices and forward zero curves, are publishedregularly by rating agencies for bonds. However, there exist no analogues directly applicable tovaluing bank loans granted to companies. Thus, bond-based variables can act as proxy variables incredit risk modeling conducted for private firms.

The main differences between rating long-term bonds and private firms are listed in Table A6.Rating bonds requires information in excess of default probabilities, primarily estimations of theloss given default (which depend on the characteristics of the bonds: secured versus unsecured,senior versus subordinate) and the issuer’s migration risk to other rating classes. Rating long-termbonds also requires qualitative assessments that go beyond financial ratio analysis, e.g. assessmentof the management. RiskCalc PDs are more volatile (reflecting changes in the financial ratiosduring the business cycle) and have a well-defined risk horizon (contrary to bond ratings, whichare valid over a nonspecific time horizon).

Table A6: The Main Differences Between RiskCalc PDs and Moody’s Long-Term BondRatings

Similarities & Differences Beteen RiskCalc Pds And Moody’s Long-Term Bond RatingsCharacteristics RiskCalc PDs Moody’s Long-Term Bond Ratings

Unit of Study Obligor Obligation and/or ObligorTime Horizon Specific, one or five years Non-specific, long termRisk Dimension One dimensional: Probability of

defaultMulti-dimensional: Probability ofdefault, severity of default & transitionrisk

Information Requirements Large, reliable, electronic datasets Robust to poor quality or missing dataVolatility High Low- maintained through the cycleCost Low HighSupport Technical Technical + Analyst contact & insightScale Continuous/Abolute 21 Risk Buckets/RelativeStructure Simple, codified analysis of few

variablesFlexible as situation may requre

Source: Moody’s RiskCalc™ For Private Companies.

For mapping purposes, RiskCalc uses standard Moody’s default tables that contain 1-year andmulti-year cumulative default rates characteristic of Moody’s rating grades. The later are defaultrates unadjusted for withdrawals observed over the 1983–1999 period.

Moody’s proposes several mapping procedures:� Use the averages of the model output per Moody’s grade: if the derivation sample contains

firms already rated by Moody’s, then one can derive an average default in each rating class. Ifthe average default rate is 0.05% for Aaa-rated firms and 0.2% for Aa-rated firms, the cut-offprobability value distinguishing these two rating classes is found between the 0.05% and 0.2%


values (the middle point, for example). Then, any firm in the prediction sample with anestimated default rate of less than 0.075% could be assigned to the Aaa rating class.

� Estimate an ordered probit model on the ratings themselves. This is a natural extension of thebinary logit or probit model. In this case the dependent variable Y takes the values 0, 1,..., n,where n is the number of rating classes. The model implicitly estimates the cutoffs for variouscategories and the coefficients of the model. The output of the model is given by default ratesfor individual firms and average default rates in different rating classes.

� Impose the condition that ratings in the sample are in the same proportion as that observedwithin the entire Moody’s rated universe. Thus, if 10% of all Moody’s-rated firms are ratedAaa, then the 10% of firms with the lowest estimated default rates are mapped to Aaa. Thisprocedure is repeated for all rating classes.

� Link the model’s default rate predictions to ratings based on standard Moody’s default ratesby rating and horizon. This mapping procedure is similar to the first one, only that in this caseMoody’s ratings in the derivation sample are not required.

A3 Numerical Implementation of the Pricing-Kernel Model

Define by L(s)>0 the pricing-kernel exponent (�=eL) as a function of the risk factor vector s. Werepresent L as

��

�

�

�)()( 0 sghhsL ,

where h0 is a constant and g� are basis functions, such as orthogonal polynomials (Tchebyshev,Hermite, etc.) or other mutually orthogonal functions. The exact choice of basis g depends on themodel for risk factor statistics. For the purposes of the present study, we are taking the risk factor

space to be the product ��

�n

k 1

]1,1[ of n identical intervals between –1 and 1 (a subset of Rn) and

functions g�

�to be indicator functions )(]

22,

23[

kmm

k ss��

�� , m=1,2,3,4, k=1,…,n. In other words,

we are looking for a particular linear combination, for each k, of the four functions of the risk

factor sk that are identically 1 on a selected sub-interval ��

��

� ��

22,

23 mm of length ½ inside [-1,1],

m=1,2,3,4, and zero elsewhere, across the risk factors k. Altogether, the pricing kernel is given by

��

� � ��

��

� ��

�

n

k m

kmm

km shh

tt e 1

4

1 22,

23

0 )(1

�

.

The basic property of the pricing kernel can be formulated in terms of the current returns on themarketable assets as

� � 1)1( 11*

��

� jt

ttt yE , j=0,1,…,N, (A2)

(with 01�ty equal to r0, the risk-free 1Y market interest rate between years t and t+1). We will

assume that jwj

t ey ��11 for j>0,

0011 i

t er ��

, and that the continuously accounted returns, wj,are linear functions of the risk factors:


Asaw �� 0 , ��

��

n

k

kjk

jj saaw1

0 .

Note that, in accordance with this definition, 000 ia � .

We also assume that the density of sk is a simple triangular function )(ss �� equal to 1+s on [-1,0], 1-s on [0,1] and zero elsewhere. Then equation (A2) can be rewritten as

1)(1

1

1

)(

1

4

1 22,

23

000

��

��

�

�

�

� ��

��

� ��n

k

sashah

n

k

jk

hj dsseeIeJj

km

mmkm

j

�

�

. (A3)

Let us define

��

��

��

�

� ��

�

� 221 1

21)(

aa eae

aaq , �

��

��

��

�

� � �

�

222 2

111)(a

eaaa

aq ,

��

��

��

�

� �� 2

23 2111)(

a

eaaa

aq , ��

��

��

��

�� a

a

eeaa

aq 224 2

11)( ,

��

��

��

0,3,2,8/3,4,1,8/11),(

jmmjaq

qj

kmjkm .

Then it can be shown that

kmh

m

jkm

jk eqI �

�

�

4

1

. (A4)

Our goal is to find coefficients h0 and kmh , m=1,2,3,4, k=1,…,n, such that conditions (A2) are

satisfied with maximum precision, e.g. by defining the quadratic error minimization criterion

� � � ��

�

� �

�

�

��

��

��

�

��

��

N

j

n

k

h

m

jkm

ahN

j

n

k

jk

ahN

j

j km

jj

eqeIeJM0

2

1

4

10

2

10

2 111 00

00

, (A5)

to be minimized with respect to parameters h0 and kmh . Formally, this objective can be treated as

that of a non-linear regression problem, with aj being observations of the n-dimensionalindependent variables, replicas of unity (for all j) – the dependent variable observations, and k

mh –the estimated coefficients.


A properly chosen minimization procedure should result in the choice of parameters kmh and h0

that makes the latter parameter a function of the former, such that the term in (A5) correspondingto j=0 is equal to zero exactly. Indeed, for any choice of k

mh , one can achieve zero error for j=0 bysetting

��

�

��

��

�

�

��

�

� ��

�

��

� ��

�

��

� �� n

k

shSh

t dsseeEH mmm

km

n

k m

kmm

km

1

1

1

)()(

)(

4

1 22

,2

31

4

1 22

,2

3

�

��

, Hih log00�� , (A6)

always reducing the overall quadratic error magnitude in (A5).

As soon as these coefficients are found, the estimated pricing kernel � can be used to calculate thecollateralized loan portfolio value in (6), Section 8. For instance, loan j maturing in year Tj>t and

having the residual service schedule � � tT

kT

kt

jj

C�

�� 1 (recall that jTT

TT FcC

j

j

j

j �� with small c beingcoupons and F the face value) has the time t-value

��

��

��

�

��

�

�

��

�

jj T

k

jkt

jkt

jktt

ktt

T

k

jkt

jktt

kttt

jt hKCKEV

11,

1,

*� . (A7)

All terms inside the conditional expectation operator in (A7) are known functions of risk factors s.The pricing kernels ��

�tt for different maturity years t+� will have the same set of coefficients k

mh(we assume a stationary distribution of risk factor variables Sk under annual evaluationfrequency). Only constants h0 are different, reflecting different discounting of individual futureperiods.

CNB Working Papers Series

9/2003 Alexis Derviz:Narcisa KadlčákováLucie Kobzová

Credit risk, systemic uncertainties and economic capitalrequirements for an artificial bank loan portfolio

8/2003 Tomáš Holub:Martin Čihák

Price convergence: What can the Balassa–Samuelson modeltell us?

7/2003 Vladimír Bezděk:Kamil DybczakAleš Krejdl

Czech fiscal policy: Introductory analysis

6/2003 Alexis Derviz: FOREX microstructure, invisible price determinants, and thecentral bank’s understanding of exchange rate formation

5/2003 Aleš Bulíř: Some exchange rates are more stable than others: short-runevidence from transition countries

4/2003 Alexis Derviz: Components of the Czech koruna risk premium in a multiple-dealer FX market

3/2003 Vladimír Benáček:Ladislav ProkopJan Á. Víšek

Determining factors of the Czech foreign trade balance:Structural Issues in Trade Creation

2/2003 Martin Čihák:Tomáš Holub

Price convergence to the EU: What do the 1999 ICP datatell us?

1/2003 Kamil Galuščák:Daniel Münich

Microfoundations of the wage inflation in the Czech Republic

4/2002 Vladislav Flek:Lenka MarkováJiří Podpiera

Sectoral productivity and real exchange rate appreciation:Much ado about nothing?

3/2002 Kateřina Šmídková:Ray BarrellDawn Holland

Estimates of fundamental real exchange rates for the five EUpre-accession countries

2/2002 Martin Hlušek: Estimating market probabilities of future interest rate changes

1/2002 Viktor Kotlán: Monetary policy and the term spread in a macro model of asmall open economy

Czech National BankEconomic Research DepartmentNa Příkopě 28, 115 03 Praha 1

Czech Republicphone: +420 2 244 12 321

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Documents

WORKING 2003 /9/ PAPER SERIES - cnb.cz